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THE UNIVERSITY 
OF ILLINOIS 
LIBRARY 


The 

Frank Hall collection 

| of arithmetics, 

presented by Professor 
H. L. Rietz of the 
University of Iowa. 


SHOT 5'3 
K63me 


MATHEMATICS LIBRARY 


The person charging this material is re- 
sponsible for its return to the library from 
which it was withdrawn on or before the 
Latest Date stamped below. 


Theft, mutilation, and underlining of books are reasons 
for disciplinary action and may result in dismissal from 
the University. 

To renew call Telephone Center, 333-8400 


UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 


AAR 9.2 REC'D 


L161—0O-1096 


THE MODEL 


HLEMENTARY 


Ree Few re: 


INCLUDING 


ORAL AND WRITTEN EXERCISES. 


BY 


ALFRED KIRK AND HENRY H. BELFIELD 
Principals of Public Schools, Chicago, 


AUTHORS OF MODEL ARITHMETIC, ETC. 


CHICAGO: 
GEO. SHERWOOD & CO. 


DORCAS AL Grr ae 


Tuts book is designed both as an introduction to the 
Model Arithmetic, and as a text-book of sufficient com- 
prehensiveness for those who do not complete the full 
grammar-school course. It has been the aim of the authors 
to present in it the accuracy of statement, clearness in 
discussion, and fullness of illustration, which are believed 
to characterize their larger work. The matter, aside from 
definitions, etc., is new, and has been tested in the school- 
room. 

The introduction will be found of great use, not only in 
developing the idea of number, but in affording material 
for constant drill in fundamental operations. 

Among the features which distinguish the book from 
many of its class, will be noticed the absence of pictures 
of common objects. No space is wasted in representing, 
by the engraver’s art, that which every teacher presents to 
the eye of the pupil. Teachers every-where recognize the 
fact that the object itself is better than the picture of it; 
that, for instance, the actual division of an apple into 
halves, quarters, etc., in the presence of a class, produces 
a clearer and more lasting conception of the idea of a frac- 
tion, than is secured by any picture, however perfect. 

The work, though elementary, is not designed as a 
plaything, but is an earnest attempt to assist in training 
the child’s mind, and in fitting him for active life. As 
such, it is commended to the public. 

K. & B. 

Chicago, April, 1876. 


CoPpYRIGHT, 1876, By Gzo. SHERWOOD & Co. 


VOde a) ML 


wes: trek TF 


ee a oe ee Se 


— {} 


f v 
KL Cane 
MATHEMATICS LIBRAK\ 


CONTENTS. 


SN CCIE ark | Chee ees el de aca oe PCat Deco can bee 
SECTION 1, Serna Pe ee oe See i os 
SECTION II, NOTATION AND NUMERATION....-- 
ee LAPP Pek LIER Ne scl i weer otoe piers he eee 
TORN SE Vr) Dt A Ore Peet. | A eu ee 
Mer etsies Ver Orrin LUN cde ecc hace see coms 
eerie Wl ead P LVS LONN © oer. Sun oo. eo ed. poe ek 
Peete Vilv UNL LELD Die o MONNY. 22. Lane. 
SECTION VIII, PROPERTIES OF NUMBERS. 
DN es Ae pa ge L024. P Mnliiples 2-2 tales iG ee 
Mameel lation <2". 2h 105 Common Multiples... -_-- 
Common Divisors___-__-_- 107 
SECTION IX, FRACTIONS. 
PISRNIONE = cee tk rks 114)" Supiraction: oo. oo. see 
REOCRUOT Ge So kn wie IG er arulaplication....— —. er. 
meric ose ee Loe? hOLIVIsiOMe: accu A ee 
SECTION X, DECIMAL FRACTIONS. 
Bien tions. C1C, soca es osc 14431 Division 3 aes nee ee 
POPEUIPOLION «hom vt oe ee 144 | Division of Decimal Mixed 
Addition and Subtraction. 146 Numbers 4-245. ee 
Miultiplication...5..0 149 
SECTION XI, DENOMINATE NUMBERS. 
Definitions and Tables.... 159 | Multiplication._......___- 
een ee TNS Pree es. a Si. 
Addition and Subtraction. 185 | Mensuration ___._________ 
EPO II A SE IGN LAGIO. Bool... Sogit de ud oacls 
(3) 


109 
110 


124 
125 


131 


151 


153 


187 
188 
190 


197 


SUGGESTIONS TO TEACHERS. 


The eight lessons that immediately follow have been arranged and 
introduced at the beginning of this book for the purpose of giving 
pupils a thorough drill in the use of the digits from 2 to 9 inclusive. 
Though for convenience of presentation the limit of the processes 
is confined to 12 times —, the teacher may extend the operations of 
addition and subtraction at will. The limit of 100, however, is all 
that is desirable with numbers less than 9. 

The, treatment of each number includes the four processes of 
addition, subtraction, multiplication and division, and recognizes the 
fact that they are intimately and naturally connected, and that one 
operation may be said to include all the others, for all the operations 
are only the comparison of numbers, one with another. They are, 
in fact, different methods of reading the same general relation of 
numbers, thus: 4+4=8 is read 4 and 4 equals 8; 2X4=8 is read 
2 times 4 equals 8; 4—2=2 is read 4 less 2 equals 2; and 4+2=2 is 
read 2 is contained in 4, twice, or 2 can be taken twice from 4, or 2 is 
one-half of 4. 

The consideration of these processes does not contemplate the use 
of the terms add, subtract, multiply or divide, as these terms will be 
more profitably discussed hereafter, but it is designed to make pupils 
familiar first with the processes themselves, and with the use of 
terms easily understood. It is assumed that pupils know how to 
count, and are familiar with the forms, names and values of the 
significant figures. While the teacher will be greatly assisted in this 
instruction by the use of lines or dots upon the board, or by a 
numeral frame, or small objects, such as grains of corn, beans, etc., 
it must not be forgotten that the memory should be made an import- 
ant factor in the mastery of these relations. Constant appeals must 
be made to the pupil’s power to gather and retain through frequent 
repetition. 

Immediately succeeding the treatment of the abstract number, a 
variety of exercises, both abstract and denominate, is given as models 
for the teacher. These exercises may be variously diversified and 
extended, according to the skill of the teacher and the necessities 
of the pupils. The lessons should be used as follows: as for in- 
stance, Lesson I; (1) reads 2, 4, 6, 8, etc., 24; (2) 1, 3, 5, etc., 23; 
(3) 2, 4, 6, etc., 24; (4) 24, 22, etc., 2; (5) 23, 21, etc., 1; (6) once, twice, 
3 times, etc., 12 times. 

(4) 


ADDITION TABLE. 
Pee el 4-91-13 eee 6 pep es-+-9 
QD | 241] 2+2 | 243 | 244 | 245 | 246] 2+7/2+8 
“of Q | 341] 8+2 | 84+3| 344 | 345/346) 3+7 
3-49 4 4t+]1}4+2/443 | 444/4+4+5/4+6 
348] 349 5 Fae | b-+2 1 5431 5441 55 
44+7|/448/449 6 6+1 | 642] 6+3 | 6+4 
§6+6|5+7]5+8 | 5+9 7 T+H1 | 742) 7+3 
6+5 |6+6 | 6+7|6+8 | 6+9 S8+1 | 8+2 
7+4|7+5|7+6 | 7+7 | 7+8 | 7-+9 Q 9+-1 
843] 844] 845] 8+6 | 8+7 | 8+8 | 8+9 10 
912/943/944|9+5 | 9+6 | 9+7 | 9+8]| 9+9 
11] 12/13} 14] 15] 16| 17] 18 
' MULTIPLICATION TABLE. 
1’s} 2’s| 3’s! 4's] 5’s! 6’s| 7's! 8’s| 9'sl10’s/11's/12’s yak 
Pa} al er 4) ey ee BV el ao ae 2 
2 Pe 2 2 2 a 2 2 2 2 2 Q 
2i 2] 4] 6] 8| 10] 19] 14] 16 | 18 | 20} 22} 24| 2 
3 3 3% 3 3 3 3 3 3 3 3 3 ; 
8). 671 8+} 12 15 118 | St. 84 1-27 | 80 182 be o 
4 4 4 4 4 4 4 4 4 4 4 4 
4 4] 8] 12] 16! 20 | 24 | 28 | 82 | 36 | 40 | 44 | 48] 4 
Ses t ebot tot oe: | Gl. 4 eto ho het <0 bee 
5 | 5110] 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50155 | 60] J 
6. (wel 6-1 26 CAC Oty .6-4)-6., eeanr ern de 
6 | 6| 12] 18 | 24 | 80 | 86-| 42 | 48; 54 | 60 | 66 | 72; 6 
yi % 7 q ré q i v§ q v4 q ¥ 
7 | %| 14] 21 | 281 85 | 42 | 491 56] 681 70/57 | 84] 7 
8 8 8 8 8 8 8 8 8 8 8 8 
& | 8] 16 | 24] 32 | 40 | 48 | 56 | 64] 72 | 80/88] 96| 8 
9 9 9 9 9 9 9 9 9 9 9 9 
9} 9118] 271 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 1108 | -9 
10 10 10 10 10 10 10 10 10 10 10 10 
10 | 10 | 20 | 30 | 4) | 50 | 60 | 70 | 80 | 90 {100 [110 1120 | ZO 
11 11 11 11 11 i il | 11 11 ne » Dh 
11 | 11 | 22 | 33 | 44 | 55 | 66 | 77] 88 | 99 |110 |121 |189 | 7Z 
12 12 12 12 12 12 12 12 Lo 12 12 13 
12 | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 |108 [120 (182 |144 | 12 


PRACTICE TABLE. 


PO ES ee ee ee Chee eek Or 
1 9 | bees Blame Sy dea, 5 
a bias Fa ee) Be OM fog Be 12 
Drie eee arenas IT sel ee ee o 
ere | Oa) aay, 1 9} 3 ih 
Des ae 55 Pe Stee s v4 8 | 4 
11 4 D Ane Died wml 1 Sofi pad 
ra 2 7112 (ile went ae eae! yaa 
rok | ea RO er On) ee re: dfs Bl 1 5 | 12 
ael OF thd BLE tO si Gale Om ane. sai 
De ES ee a ee pV el beaey On 8a 
LOSS Ga So Ost 10 Hie tate: 
diye bees RO Ra eG oh erie! a De i es 5 


The above table is to be used as follows: 


1. Apprrion. Let the pupil add to each number in a given column 
a number announced by the teacher, the pupil stating the 
successive swms only. Thus, the teacher will say, ‘Column A, 
add 5; the pupil selected answers as rapidly as possible, “ 6, 8, 
10, 12, 14,” etc. 

2, Suprraction. Let the pupil subtract from a given number each 
number in a given column. Thus, the teacher announces 
“Column B, subtract from 17.” The pupil subtracts each 
number, mentally, and states the result only, “8, 12, 14, 16,” ete. 

3. Mourrerication. The teacher selects a column and a mul- 
tiplier. The pupil announces the product, using each number 
as a multiplicand. Thus, “C by 5,” results in 60, “5, 15, 45, ete. 

4. Diviston. The teacher selects a column and announces a 
dividend; the pupil states the quotient, using each number as a 
divisor. It is not necessary that the dividend selected be a 
multiple of any of the divisors. Thus, the aay may 
announce “ D, dividend 20;” the pupil responds 2 24, 5, 20, 28, 
6%, etc. 

In all of these exercises accuracy should be insisted upon from the 
beginning; rapidity should not be expected at first, but will result 

from constant practice. (6) 


MODEL 


ELEMENTARY ARITHMETIC. 


ENS Roe Gal ON 


LESSON I. 
DEVELOPMENT OF NUMBERS BY 2’s. 


Z. 24-24-24-24-24-24-24-242424212=? 

Q. LF VABHVHVARHAFRARFRHR42=? 

Pf) ee ae a a eee, OSE 
ee ee ea uO ee ed Kee LO Sat 
1 edb — hd Way Sam o 


4, 24—2—2—2—2—A2—2—-2—2—--2--2—-2=? 

H. 23—2%—2%—2—-2—2—-2—2—-2—-2—-2-2=? 

6. 2+2=? 422=? 622=? 8=2=? 10+2=? 
12+2=? 14+2=? 16+2=? 18+2=? 20+2=? 
22—-2=? 24+2=? 


7. 4 is 2 more than what number? 2 less than what 
number? 
8. 15 is 2 more than what number? 2 less than what 
number? 
9. 12 is twice what number? 22 is twice what number? 
1O. What number should be doubled to obtain 24? 
What number is contained in 18 twice? 
11. What number is 2 more than 18? 2 less than 16? 
12. What number should be added to 12 to obtain 14? 


2 can be taken from 6 how many times? 
(7) 


8 ELEMENTARY ARITHMETIU. 


18. 6 is one half what number? 8 is one half what 
number? 

14. 11 is one half what number? 18 is twice what 
number? 

15. 12 is double what number? Of what number is 7 
one half? 

16. 9 is one half what number? What five equal 
numbers equal 10? 

17. 13 is equal to what six equal numbers and one 
more? 8 times 2 and 1 more equals what number? 

iS, 11 times 2 and 1 more equals what number? 


19. A boy had 16 cents, and gave 2 cents to his sister. 
How many cents had he left? | 

20, Alfred had 22 marbles, and found 2 more. How 
many did he then have? 

21. If one yard of cloth costs 6 dollars, how much will 
2 yards cost? 

22. John has 9 chestnuts, and his sister has twice as 
many. How many has his sister? 

23. If one top costs 2 cents, how many tops may be 
bought for 16 cents? 

24. How many two-cent stamps may be bought for 24 
cents? 

25. A girl paid 22 cents for a book, and one half as 
much for a slate. How much did she pay for the slate? 

26. Paid 9 cents for a lead pencil, and twice as much 
and 1 cent more for some paper. How much was paid for 


the paper? 


INTRODUCTION. g 


LESSON TT. 


DEVELOPMENT OF NUMBERS BY 3's. 


pats +388 45-184- 3348-53 34+ 3? 

2 2434343438434343434343+3=? 

do 14+34384343434343434+3+4+34+3=? 

Pee cate ON oat oo eo Fe Oa OD KOSET 
PORES CH as and dial Dar) came ga rs nigel A as 
Bt a fe ad Dog Pome 

Oho obo — 3 3-6 0 oO 

fPema0e =o 5 SO Oe 

fot 30 — 5 — > — 5 —o—o--o o-oo 

8. §+-3=? ee ee een lee = oe 


18+3=? 21+3=? 2423=? 27+3=? 30+3=? 


9. 12is 3 more than what number? 3 less than what 
number? 

10. 26is 3 more than what number? 3 less than what 
number? 

11. 6 times 3 equals what number? 9 times 3 equals 
what number? 

12. 15 is 3 times what number? 27 is 3 times what 
number? 

13. What number is contained 3 times in 24? What 
number is equal to 3 times 7? , 

14. What number is 3 less than 22? 3 more than 28? 

15. 36 is 3 times what number? 3 can be taken from 
12 how many times? 


10 ELEMENTARY ARITHMETIC. 


16. Of what number is 8 one third? 6 is one third of 
what number? 

17. What 2 equal numbers are contained in 6? What 
4 equal numbers equal 12? 

18. times 3 and 2 more equals what number? 9 times 
3 and one more equals what number? 

19. What is one of the 6 equal numbers contained in 
18? 3 is contained in 27 how many times? 


20. Henry had 25 dollars, and earned 3 dollars more. 
How many did he then have? 

21. A girl having 31 cents, spent 3 cents for pins. How 
many cents did she have left? 

22. One ton of coal costs 8 dollars. How much will 3 
tons cost at the same rate? 

23, James had 12 pigeons, and his father had 3 times 
as many. How many pigeons had his father? 

24. One yard of ribbon costs 3 cents. At the same 
rate, how many yards may be bought for 18 cents? 

25. How many yards of cloth, at 3 dollars a yard, may 
be bought for 36 dollars? 

26, One third of 24 dollars was paid for a hat. How 
“ much did it cost? 

27. A vest cost 9 dollars, and a coat 3 times as much, 
and 2 dollars more. How much did the coat cost? 


INTRODUCTION. 11 


LESSON III. 


DEVELOPMENT OF NUMBERS BY 4’s. 


2. 44-44-44-44-444441414141441414—) 
2 $4+444444-44444444414144444=? 
3b. 2+4444-44-41444444444444414=? 
4A. 14444444444444444444444=? 
eet he oh A OAK OD Ka? 
Cee tee oe ee Oe 4 1 Os eee 
Be he ae 
G6. 48—4—4—4—4—4—4—4—4—4-—4-—4=? 
7 47—4—4—4—4—4—4—4—4_-4-—-4--4=? 
&. 46—4—4—4—4—4—4—4-—4—4-—4-4=? 
9. 46—4—4—4—4—4—4-—4-—4—4-—4-4=? 
f0, 424=—7 $= 4=7 12=4=—? 16-4=? 20-4=? 
; 24-4=? 28-4=? 32+4=? 36+4=—? 40-4=? 
44--4—=? 48-4=P?P 


Zl. 20is 4 more than what number? 4 less than what 
- number? 

712. 38 is 4 more than what number? 4 less than what 
number? 

13. 28 is four times what number? 48 is 4 times what 
number? 

14. What number is contained 4 times in 48? 4 times 
in 36? 

15. What number is 4 more than 27? 4 less that 45? 

16. What number should be added to 33 to obtain 37? 
4 can be taken from 20 how many times? 


12 ELEMENTARY ARITHMETIC. 


17. 3 is one fourth what number? 8 is one fourth 
what number? 

18. Of what number is 7 one-fourth? 6 is one fourth . 
what number? 

19. What number should be added to 4 times 7 to 
obtain 82? 4 times 6 and 3 more equals what number? 

20. 9 times 4 and 2 more equals what number? 4 is 
contained in 48-how many times? 


21. A man had 35 cents, and gave 4 of them for some 
apples. How many had he left? 

22. Grace had 39 nuts, and found 4 more. How many 
had she then? 

23. How much will 4 oranges cost, if 1 orange costs 9 
cents? 

24, James has 12 dollars, and his father has 4 times as 
many. How many has his father? 

25. If one hat is worth 4 doliars, how many hats may 
be bought for 28 dollars? 

26. Julia is one fourth as old as her mother, and her 
mother is 36 years old. How old is Julia? 

27. A boy had 12 cents, and his sister had 4 times as 
many. How many cents had his sister? 

28. <A girl is 9 years old, and her father is 4 times as 
old, and 3 years more. How old is her father? 

29. A person had 24 dollars, and put it into piles of 4 
dollars each. How many piles were there? 

30. Anna bought 8 pears at 4 cents each, and had 3 
cents left. How much money had she at first? 


INTRODUCTION. 18 


LESSON IV. 


DEVELOPMENT OF NUMBERS BY 5's. 


We 6--5-+-5--6 4-5 4-5 -+-6-45-+4-5-+5-+54-5=? 

Rg eet gee ie ye ae 

Me 25 1-5-5 +--+ 54-5-++6-+-5-15-+5--5=? 

4. 2+5415+5+54+5+5+54+5+5+4+5+4+5=? 

5. 14+54+5+54+54+54+54+54+5+5+45+45=? 

See ye a ay el a re Sk tie 
Gt hee os te ED On BU a et xX Oe 
tie Tel eee at 

7% 60—5—d5—5—5—5—5—5—5—5—5—5=? 

& 59—5—d—d—5—d5d—d—5d—5—5—d)—5=? 

fA RS 5 Re et ee a ee te 

10 oD) 0 — 0a O05 OS 
TI. 56—5—5—5—5—5—5—5D—5—5 —~-5—5=? 

12. 5+5=? 10+5=? 15+5=? 20+5=? 25+5=? 
30+5=? 35+5=? 40+5=? 45+5=? 50+5=? 
55h 7- 60=-h =? 


18, 251s 5 greater than what number? 5 less than what 
number? 

74. 40 is 5 more than what number? 5 less than what 
number? 

15. 35 is 5 times what number? 60 is 5 times what 
number? 

16. What number is contained 5 times in 45? 5 times 
in 30? 

17. What number is 5 less than 43? 5 more than 53? 


14 ELEMENTARY ARITHMETIC. 


18. What number is 5 less than 39? What number 
should be added to 54 to obtain 59? 

19, How many times can 5 be taken from 30? 6 is one- 
fifth of what number? 

20. 9 is one fifth of what number? Of what number 
is 10 one fifth? 

21. 12 is one fifth of what number? What number 
should be added to 5 times 8 to obtain 45? 

22. 85 times 7 and 4 equals what number? What four 
equal numbers equal 20? 

23. What five equal numbers and 3 more equal 28? 
What nine equal numbers and 4 more equal 49? 

24. 36 is equal to what seven equal numbers and one 
more? 27 is equal to what five equal numbers and 2 more? 


25. Lucy took to market 50 cents, and spent 5 cents for 
lettuce. How many cents had she left? 

26. Howard owned 35 cents, and his mother gave him 5 
cents more. How many cents did he then have? 

27. Tfoné yard of velvet costs 7 dollars, how much will 
5 yards cost? 

28. James is 9 years old, and his mother is 5 times as 
old. How old is the mother? 

29. A gentleman paid 11 dollars for a gun, and five 
times as much and 4 dollars more for a wagon. How much 
was the wagon worth? 

30. If a boy can gather 5 quarts of nuts in one day, in 
how many days can he gather 40 quarts, at the same rate? 

31. How many quarts of berries, at 5 cents a quart, 
may be bought for 55 cents? 

32, A lady paid one fifth of 60 dollars for a shawl. 
How much did the shaw] cost? 

33. How many five-cent pieces are equal to 45 cents? 


INTRODUCTION. 15 


iis DN 


DEVELOPMENT OF NUMBERS BY 6's. 


6+6+6+6+6+6+46+46+6+6+6+4+6=? 
5+6+6+6+6+6+6+4+6+6+6+6+6=? 
44+6+6+46164+6+46+646+4+646+6=? 
3-+6146464614646+6+4+6+6+64+6=? 
2+646464646+4646464646+6=? 
14+64+6+46+6+46+6+4+6+4+6+6+46+6=? 
Pie hee oe or a OSS ke OS tO COME 
Pee ee a bee, OX 6 TP 10 xX oe 
Pee ie ele Grr 
5. %72—6—6—6—6—6—6—6 


MOV Se we mM 


TZ. 6+6=? 12+6=? 18+6=? 24-6=? 30-6=? 
Bo Gat ee 46 od eh) 50-62 
Bh=-6 =. fn b . 


15. 36 1s 6 greater than what number? 6 less than what 
number? 

76. 58 is six more than what number? 6 less than what 
number? . 

17. 42 is 6 times what number? 66 is 6 times what 
number? 


16 ELEMENTARY ARITHMETIC. 


18. What number is contained 6 times in 48? 6 times 
in 72? 

19, What number is 6 less than 65? 6 more than 47? 

20, What number is 6 less than 37? By what number 
should 43 be increased to obtain 49? 

21. How many times can 6 be taken from 24? 8 is one 
sixth of what number? 

22. 5isone sixth of what number? Of whit number 
is 9 one sixth? 

28. 12 is one sixth of what number? By what number 
should 6 times 10 be increased to obtain 66? 

24. 6 times 7 and 5 more equals what number? What 
three equal numbers equal 18? 

25. What four equal numbers and 3 more atl 27? 
50 is equal to what 8 equal numbers and 2 more? 

26, 57 is equal to how many 6’s and 3 more? 70 is 
equal to how many 6’s and 4 more? 


27. Frank received for a gift 55 cents, and expended 6 
cents for an orange. How many cents had he left? 

28. Sarah earned 48 cents by sewing, and 6 cents more 
by knitting. How much did she earn in all? 

29. If one barrel of flour is worth 8 dollars, how many 
dollars are 6 barrels worth? 

30. A boy can walk 9 rods a minute, and a dog can run 
6 times as far in the same time. How far can the dog run 
in oue minute? 

31. 72 dollars is 6 times what a gentleman paid for a 
sheep. How much was paid for the sheep? 

32, One sixth of 48 dollars is the price of one yard of 
velvet. How much is the velvet worth? 

33. A paid 7 dollars for a vest, and 6 times as much and 
5 dollars more for a coat. How much did they both cost 
him? 

34. How much is paid for a cow that costs 11 times 6 
dollars and 4 dollars more? 


INTRODUCTION. 17 


LESSON VI. 


DEVELOPMENT OF NUMBERS BY 7s. 


17. 56 is 7 more than what number? 7 less than what 
~ number? 
18. %8is 7 greater than what number? 7 less than what 
number? 
19. 49 is 7 times what number? 63 is 7 times what 
number? 
20. What number is contained 7 times in 56? In 84? 


* 


DL, CTIA T AT AT AT H74-74-74747=? 
2 GATATATHTAIATF7I47474747=? 
B. BATHTATETATATLIFTAI AI 47 =? 
fe ALTHIATATATATATLTAI FIT 7=? 
| B BATHTATHTATAT ETAT AT HITS 
6. QATATATATATATATAIAIAI4I= 
7 VATATHTATHTATAT AT ETAT I= 
236, Me Bet 0g nme ERTS eet de: Sd ome g hed 
oo ene IW Bema astra ge ait Dv al eat i LU geen 
hip ee 250% 
eee ey a 
IO. 83—7—7—7—7—7—7-—7-—7-—7-—7-—7=? 
11. 82—7—7—7-—7—7—7-—7—7-—7—7 -—7=? 
12. 81—7—7—7—7—7-—7-—7—7—-7-7-7=? 
18. 80—7—7—7—7—7—7-—7-—7-7-7-7=? 
LZ. (9—7-—7—T-—7-—7-—7-—7-7-7-7-7=? 
15. %8—7—7—7—7-—7-—7-—7-—7-—7-—7-7=? 
erent) — pe iad 7 al on OS ae Rh ee? 
43-7 =? 4927=? 56-7=—? 638-7=? TO+7=? 
17+7=? eset | 


18 ELEMENTARY ARITHMETIC. 


21. What number is 7 less than 75? 7 greater than 64? 

22. By what number should 65 be increased to obtain 
72? How many times can 7 be taken from 35? 

23, 6isone seventh of whatnumber? 9 is one seventh 
of what number? 

24. 121s one seventh of what number? By what num- 
ber should 8 times 7 be increased to obtain 63? 

25. % times 7 and 6 more equals what number? What 
4 equal numbers and 4 more equal 32? 

26. What 6 equal numbers equal 42? What 5 equal 
numbers and 3 more equal 38? 

27. V5 is equal to what 10 equal numbers'and 5 more? 
67 is equal to how many 7’s and 4 more? 

28. 83 is equal to how many 7’s and 6 more? One 
seventh of 84 equals what number? 


29. Clara bought a paper of 72 pins, and gave away 7. 
How many pins had she left? 

30, 59 dollars is 7 dollars less than what was paid for a 
sleigh. How much was paid for the sleigh? 

31, If one bushel of chestnuts is worth 7 dollars, how 
much are 9 bushels worth? 

32. How many barrels of flour, at 7 dollars a barrel, 
may be bought for 56 dollars? 

33. How much is paid for a quantity of corn that costs 
11 times 7 dollars and 5 dollars more? 

34, 63 acres is 7 times what a farmer has planted in 
wheat. How many acres of wheat has he? 

55, One seventh of 77 years is George’s age. How old 
is George? 

36. There are 12 inches in one foot. How many inches 
are there in a line 7 feet long? 

37. How much is the rent of a house a month for which 
there is paid 9 times 7 dollars and 6 dollars more? 

38. 61 dollars is 5 dollars more than what 7 tons of coal 
cost. What is the price of one ton? 


ES Se DEO RN Se Se. 


19. 


number? 


INTRODUCTION. 


LESSON VII. 


2X 


i 


DEVELOPMENT OF NUMBERS BY 8's. 


84+8+8+8+48+48+4848+48+48+8+8=? 
7+84+8+48+48+8+848+8+8+8+8=? 
6+8+8+48+48+48+4848+8+8+8+8=? 
5+8+4+8+848+48+48+48+48+8-+-8+4+8=? 
44+84+84+848+848+48+4848+8+8=? 
3+8+8+4+84+848+8+48+48+48+48+8=? 
24+848+4+8+4+8+48+8+848+48+48+48=? 
1+84+8+4+8+4+8+48+48+4848+8+8+8=? 

Ls? 


19 


Ox = fiat BaP so Boel 

bt Sea Oe oer Le 
—8—8—8—8—8-—8=1 
—8—8—8—8—8—8=? 
—8—8—8—8~—8-~—8=? 
—8—8—8—8~—8-—8=? 
—8—8—8—8~—8-—8=? 
—8—8—8—8—8—8=? 
—8—8—8—8—8-—8=? 
—8—8—8—8—8—8=? 
24=8=? 32—-8=? 40+8=? 
64+8=? 72+—8=? 80+8=? 


48 is 8 more than what number? 8 less than what 


20. 57 is 8 greater than what number? 8 less than what 


number? 


20 ELEMENTARY ARITHMETIC. 


21. 5 times 8 equals what number? 8 times 9 equals 
what number? 

22. What number is contained 8 times in 56? In 64? 

23. By what number should 53 be increased to obtain 
61? What number is 8 more than 49? 

24. What number is 8 less than 77? 96 equals 8 times 
what number? 

25, What number equals 8 times 7 and 6 more? What 
number equals 7 more than 8 times 11? 

26. How many times can 8 be taken from 48? 9 equals 
one eighth of what number? 

27. 12 equals one eighth of what number? Of what 
number is 8 one eighth? 

28. One eighth of 48 is 3 less than one eighth of what 
number? 6 times 8 and 7 more equals what number? 

29, What four equal numbers and 5 more equal 37? 
What 6 equal numbers are contained in 48? 

30. What 8 equal numbers and 5 more equal 69? What 
7 equal numbers are 6 less than 62? 


31. Horace in one day earned 56 cents, and gave 8 of 
them for a lead pencil. How much had he left? 

32, Julia planted 65 seeds, and Mary planted 8 less than 
Julia. How many did Mary plant? 

33. There are 9 square feet in one square yard. How 
many square feet are there in 8 square yards? 


\ 


34. Ifaperson work 8 hours each day, how many hours 
will he work in 12 days? 

35. A boy was asked how many marbles he had, and 
replied that if he had 8 times as many, he would have 96. 
How many marbles had he? 

36. If a ship sails 8 miles an hour, in what time, at the 
same rate, will she sail 88 miles? 

37. One-eighth of 72 dollars is what a gentleman paid 
for 2 tickets to the opera. How much did they cost? 


INTRODUCTION. 21 


ete N yt Ls 


DEVELOPMENT OF NUMBERS BY 9's. 


1. 94+94+9+49494949494949+4+9+49=? 

2. 84+-949494949494949494949=? 

3 %74+949494949494949494949=? 

4. 6+94+94194949+49494949+949=? 

5d. 64+-949494949494949494949=? 

6. 44-949494949494949494949=? 

7. 3+949494949491949494919=? 

§& 24+94+94949494941949494949=? 

9. 14+949+49+4949494949494949=? 

PO don pe se Oar wt Os A OS Ph SO 2 
ee ESF hae yt Lome ae Sreks Fam 4 Be ahs Pama g i NWT Bone 
[TST fame Ha RA! Pg 

11. 108—9—9—9—9—9—9—9—9—9—9—9=? 

12. 107—9—9—9—9—9—9—9—9—9—9—9=? 

13. 106—9—9—9—9—9—9—9—9—9-—9—9=? 

feet 9-- 02.95 9-9. 99 __9 9-9 

feed = O.- 0.9 =) 9 6 9a =o. oF 

16. 103—9—9—9—9—9—9—9—9—9—9—9=? 

17. 102—9—9—9—9—9—9—9—9—9—9—9=? 

18, 101—9—9—9—9—9—9—9—9—9—9—9=? 

19. 100—9—9—9—9—9—9—D—9—9—9—9=? 

20, 9=9=7) 18+9=? 27-9=7'°36+9=7 45-97? 
p4--O-=-? 63--0=fiie v=? SI=9=—?, 90-9? 
99--9=? 108+9=? 


21. 45 is 9 more than what number ? 9 less than what 
unmber? 


| 


php ELEMENTARY ARITHMETIC. 


22. %6 is 9 greater than what number? 9 less than 
what number? 

23, 6 times 9 equals what number? 9 times 7 equals 
what number? 

24. What number is contained 9 times in 72? In 108? 

25. By what number should 58 be increased to obtain | 
67? What number is 9 more than 85? | 

26. What number is 9 less than 97? 81 equals 9 times 
what number? 

27. What number equals 9 times 6 and 7 more? How 
many times can 9 be taken from 54? 

28. 8 equals one ninth of what number? 11 equals 
one ninth of what number? 

29. Of what number is 10 one ninth? One ninth of 
81 equals what number? 

350, 7 times 9 equals * less than what number? What 
5 equal numbers are contained in 45? 

31, What 7 equal numbers and 6 more equal 69? What 
8 equal numbers are 5 less than 77? 

32, What 9 equal numbers and 7 more equal 88? One 
ninth of 45 is 4 less than one ninth of what number? 


oo. James has 69 pennies in one box, and 9 pennies 
in another. How many pennies are there in the two 
‘boxes? 

34. Mary had 76 pinks in her garden, and 9 of them 
were destroyed. How many had she left? 

3). A farmer has 12 sheep in the barn, and 9 times as 
many in the field. How many sheep are in the field? 

36. How many more are there in the field than in the 
barn? 

37. Of the pupils of a school 6 were absent, and 9 
times as many were present. How many were present? 
How many belonged to the school? 

38. At 9 cents a pound, how much will 8 pounds of 
sugar cost? 


INTRODUCTION. 23 


39. At 9 shillings a bushel, how many bushels of potatoes 
may be bought for 99 shillings? 

40. If 9 men can pick 108 bushels of apples in one day, 
how many bushels can one man pick? 

41. What is one yard of velvet worth if 9 yards are 
worth 72 dollars? 

42. One ninth of 63 dollars is what John paid for a hat. 


What did the hat cost? 


43. Jane has 11 dollars, and her father has 9 times as 
much and 8 dollars more. How many dollars has her 
father? How many dollars have both? 

44. Jennie’s grandfather is 72 years of age, and she 
is one ninth as old. How old is Jennie? In how many 
more years will she be 17 years old? 

45. A gentleman spent 45 dollars in purchase of cloth 
at 5 dollars a yard. How many yards did he buy? How 
many dollars had he left? 

46. At 6 dollars a ton, how many tons of coal can I buy 
if I have 76 dollars? How many dollars will I have left? 

47, One seventh of 84 dollars is 3 times the cost of my 
chair. How much did my chair cost? 

45. Sold 12 barrels of flour, worth 8 dollars a barrel, for 
89 dollars. How many dollars did I lose? 

49, Wow much more than 96 dollars should a gentleman 
have in order to buy 12 cords of wood at 9 dollars a cord? 

50. A boy walked 12 days, at the rate of 9 miles a day. 
How many miles farther should he walk so as to travel 120 
miles? 

51. A gentleman earned 9 dollars a day for 8 days, and 
spent 8 dollars a day for 8 days. How much has he left? 

52, Bought 8 yards of cloth at 12 dollars a yard, and 
gave in payment 10 ten-dollar bills. How much change 
should I receive? 


24 ELEMENTARY ARITHMETIC. 


[PDs CENA T o8 BG 


PRACTICE EXERCISES. 


This lesson should be studied as follows: Use first the sign +, as 
11-+42=18, etc.; then the sign —, as 11—2=9, etc.; afterwards both 


signs, as 1142—18 or 9. 


Go NSD Oc & So ON 


11+2=? 
11+3=? 
1144=? 
1i+o=7 
11+6=? 
11+%=? 
11+8=? 
114+9=? 


1242=? 
124-3? 
12+4+4=? 
12+5=? 
124+6=? 
1247=? 
124+8=? 
12+9=? 


1384-27 
13+3=? 
138+4=? 
138-57 
134+6=? 
13+7=? 
13+8=? 
18+9=? 


21+2=? 
21+3=? 
214+4=? 
214+5=? 
214+6=? 
214+7=? 
21+8=? 
214+9=? 


224+2=? 
22+3=? 
22+4=? 
22+5=? 
22+6=? 
22+7=? 
22+8=? 
22+9=? 


2342=? 
23+3=? 
20+4=? 
23+5=? 
23+7=? 
23+8=? 
239+9=? 


81+2=? 
3143=? 
31+4=? 
314+5=? 
31+6=? 
S144 =? 
3148=? 
31+9=? 


B24+2=? 
3824+3==? 
d2+4=? 
d2+5=? 
o2+6=? 
B24+7=? 
3824+8=? 
B24+9=? 


338+2=? 
Bots Otor 
838+4=? 
dd +0=37 
33+6=? 
sd+7=? 
33+58=? 
338+9=? 


AL 4+2=? 
Al 5? 
41+4=? 
414+5=? 
41+6=? 
Al +7 ==? 
41+8=? 
414+9=? 


4242=? 
424+3=? 
42+4+4=? 
4245=? 
4246=? 
424+7=? 
424+8=? 
4249=? 


A$+2=? 
434+3=? 
A3+4=? 
434+5=:? 
43+6=? 
434+7=? 
434+8=? 
434+9=? 


26. 


14+2=? 
144+3=? 
144+4=? 
14+5=? 
144+6=? 
144+7=? 
144+8=? 
144+9=? 


15+2=1 
154+3=? 
15+4=? 
15+5=? 
15+6=? 
18+7=? 
15+8=? 


we 10+-9=? 


16+2=? 
16+-3=? 
16+4=? 
16-567 
16+6=? 
16+7=? 


. 164+8=? 


164+9=? 


174+2=? 
17+3=? 
17+4=? 
17+5=? 
17+6=? 
WWtiv%=? 
17+8=? 
17+9=? 
A2 


INTRODUCTION. 


2442=? 
244+3=? 
24+4=? 
24+5=? 
24+6=? 
244+7=? 
244+8=? 
244+9=—? 


2594+2=? 
20+3=? 


20+4=? . 


25+5=? 
25+6—=? 
20+7=? 
25+8=? 
25+9=? 


26+3=? 
26+4=? 
26+5=? 


26462? 


264+7=? 
264+8=? 
264+9=F 


R27 +2=? 
27+3=? 
27+4=? 
27+5=? 
27+6=? 
27+7=? 
27+8=? 
27+9=? 


34+2=? 
34+3=? 
344+4=? 
34+5=? 
344+6=—? 
d4+7=? 
34+8=? 
3d44+9=? 


vo +2=? 
39+3=? 
oo +4=? 
so9+5=? 
30 +6=? 
s0+7=? 
35+8=? 
so+9=? 


36+2=? 
36+3=? 
36+4=? 
396 +5=? 
36+6—=? 
36+7=? 
36+8=? 
36+9=? 


37 +2=? 
a/+3=? 
87+4=? 
si +5=? 
a7 +6=? 
dT 47>? 
37+8=? 
37+9=? 


4442-> ? 
444+ 3:>-? 
444+4=? 
4445=? 
444+6=? 
444%=? 
444+8=? 
444+9=? 


46+7=? 
46+8=? 
46+9=? 


AV4+2=? 
474+3=—? 
47+4=? 
4A74+5=? 
47+6=? 
AV+7=? 
47+8=? 
474+9=? 


etc. 


25 . 


26 


57, 
58. 
59. 
60. 
61. 
62. 
68. 
64. 


65. 
66. 
67. 
68. 
69. 
70. 
ae 
V2. 


KLEMENTARY ARITHMETIC. 


18+2=? 
18+3=? 
18+4=? 
18+5=? 
18+6=? 
184+7=? 
18+8=? 
18+9=? 


194+2=? 
19+3=? 
19+4=? 
19+5=? 
19+6=? 
19+7=? 
194+8=? 
194+9=? 


28+2=? 
28+3=? 
28+4=? 
28+5=? 
284+6=? 
28+7=? 
28+9=? 


2942=? 
294+3=? 
294+4=? 
29+5=? 
294+6=? 
29+4+7=? 
294+8=? 
29+9=? 


384+2=? 
384+3=? 
38+4=? 
a8+d5=? 
38+6=? 
s8+7=? 
38+8=? 
d8+9=? 


3894+2=? 
394+3=? 
394+4=? 
394+5=? 
39+6=? 
39+7=? 
394+8=? 
394+9=? 


48+4+2=? 
48+3=? 
481+4=? 
48+5=? 
48+6=? 
48+7=? 
484+8=? 
48+9=? 


49+4+2=? 
494+3=? 
49+4=? 
49+5=? 
49+6=? 
A94+-%=7 
494+8=? 
494+9=? 


ae 


DEFINITIONS. BT 


Py Ce OLN: aE. 


ete TL OFN: Sy 


Article 1. A Unit is a single thing, or one of any 
kind. 


Thus, one, one hat, one gallon, one year. 


One denotes a single thing; one and one more is called 
Two; two and one more is called Three; and thus, by 
the successive additions of one, are obtained the several 
numbers one, two, three, four, five, six, seven, eight, nine. 

2, A Number is an expression of one or more units. 

Thus, one, four, six miles, fifty dollars, one hundred, are numbers. 

3. Arithmetic is that branch of mathematics which 
treats of numbers and their applications. 

Hence, all forms of counting and business operations involving 
the use of numbers belong to Arithmetic. 

4, A Denominate Number is a number whose 
kind of unit is named. 

Thus, 2 apples, 5 bushels, 30 yards, are denominate numbers. 

5. An Abstract Number is a number whose kind of 
unit is not named. 


Thus, 4, 7, 40, 100, are abstract numbers. 


6. The Unit or Unit Value of a number is one of 
that number. 


Thus, i foot is the unit value of 5 feet or 10 feet; 1 is the unit value 
of 5, 10 or 50. 


28 ELEMENTARY ARITHMETIC. 


, Like Numbers are numbers that have equal unit 
values. 

Thus, 2 miles and 8 miles are like numbers, because they have 
equal unit values, 1 mile, 1 mile. 


8. Unlike Numbers are numbers that have different 
unit values. 

Thus, 5 boys, 10 apples, 20, 30 dollars, are unlike numbers, because 
their unit values, 1 boy, 1 apple, 1, 1 dollar, are not alike. 

9. Units are of the same kind, when they are either 
abstract or of the same denomination. 

Thus, 2, 4, 10, 50, express units of the same kind. 

So, 2 feet, 4 feet, 10 feet, 50 feet. 


10. Units are of different kinds cree they are of dif- 
ferent denominations. 


Thus, 3 dollars, 4 gallons, 2 quarts, express units of different kinas. 

11. Simple Numbers are numbers that express 
units of the same kind. 

Thus, 4, 6, 12, 25, 48, are simple numbers. 

So, 3 years, 40 years, 100 years. Hence, 

Simple numbers are either abstract or denominate. 

12. Compound Numbers are numbers that, re- 
garded as one quantity, express units of different kinds. 

Thus, 12 feet 4 inches, 8 pounds 6 ounces, are compound numbers. 
Hence, 

Compound numbers are always denominate. 


13. An Integer or Integral Mumber is a number 
that expresses one or more whole units. 


Thus, 7, 16 pears, 100, are integers. 


44, Figures are,characters used to represent num- 
bers. 


——— rr rr—C 


NOTATION AND NUMERATION. 29 


Seer Owe, Dk. 


NOTATION AND NUMERATION. 


Art. 15. Notation is a method of representing 
numbers by written characters. There are two methods 
of notation in ordinary use. 

1. The Roman Notation, which employs /etters. 

2. The Arabic Notation, which employs figures. 

The letters employed in the Roman Notation are the 
seven capital letters, I, V, X, L, C, D, M. 

I=—1. V=5. X=10. L=50. C=100. D=500. M= 
1000. 

Other values are represented by certain combinations of 
these letters in accordance with the following 


PRINCIPLES. 


1. Ifa letter is repeated, or if a letter or combination of 
letters of less value follows a letter of .greater value, the 
sum of the:values is the value of the combination. 

SOUS we secs a oles V AO LO G00, ¢ XTX =19: 

2. If a letter of greater value follows a letter of less 


value, the difference of the two values is the value of the 
combination. 


Thus, IX=9. XL=40. CD=400. 

3. A dash (—) placed over a letter or combination of 
letters, gives it a thousand-fold value. 

Thus, I=one thousand. XX=twenty thousand. 


30 ELEMENTARY ARITHMETIC. 


16. The figures employed in the Arabic Notation are 
the ten figures, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. 

Other values are represented by these figures, in certain 
simple combinations, in accordance with principles explained 
in Art.17. 

The first of these figures is called zero, naught, or cipher, 
and represents no value, for its use implies the absence of 
number. The remaining nine figures are called digits or 
significant figures, representing one, two, three, etc., units. 

17. Orders of Figures are represented in the 
following manner. A figure in the first or right hand place 
isa figure of the first order; in the second place, a figure 
of the second order; in the third place, a figure of the third 
order: and so on, for all successive orders. 

Thus, in the number 4765, the 5 is a figure of the first order; the 6 


is a figure of the second order; the 7 is a figure of the third order; 
and the 4 is a figure of the fourth order. 


PRINCIPLES. 


1. In counting, one, or a single thing, is the basis of all 
numbers, and is called wnt, or first order. 

2. Ten single things, or units, make one group or col- 
lection, called tens, or second order. 

3. Ten of the collections called tens make one group or 
collection called hundreds, or third order. 

4. Ten of the collections called hundreds make one 
group or collection called thousands, or fourth order. And 
similarly, ten collections of any order make one collection 
of an order of the next larger denomination. The first 
ten of these orders are named as follows: Units, Tens, 
Hundreds, Thousands, Ten-thousands, Hundred-thousands, 
Millions, Ten-millions, Hundred-millions, Billions. 

The first three of these orders, units, tens and hundreds, 
form a group called the Peried of Units; the second 
three, in ike manner, form a group called the Period of 


NOTATION AND NUMEKATION. 31 


Thousands, and in like manner, the successive higher 
periods, of Millions, Biltions, Trillions, etc. 

Thus, 345628 is a number composed of 345 thousands, 628 units, 
and is read 345 thousand 628. 

18. Figures, it is plain, may therefore represent two 
values, viz.: Absolute and Local. 

Absolute Value is represented by a figure, or by a 
combination of figures, standing alone. 

Thus, 4, 6, 8, 24, 149, represent absolute value. 

Local Value of figures, whether represented by a 
figuro or combination of figures, is the value as determined 
by the place they occupy. 

Thus, in 576, the 6 of the first order, the 7 of the second, and the 
5 of the third order, each represent a local value. 

So, the 76 and the 57. 

19. Numeration is the method of reading numbers. 

Thus, 2786 is read two thousand, seven hundred, eighty-six: or, 
six units, eight tens (eighty), seven hundreds, and two thousands. 

20. Rule for Notation.— Begin at the highest or 
lowest order, and write in each successive order the figures 
belonging to that order, filling the vacant orders with ciphers. 

Rule for Numeration.— Begin at the highest or 
lowest order, and read the successive orders of figures, giving 
the name of each period, except the period of units. 


21. TABLE OF NOTATION AND NUMERATION. 


5th 4th 3d 2d 1st 
Period. Period. Period. Period. Period. 


NAMES . 2 
Z of ‘i z 
OF S a 5 5 
— ° cian 3 a 
— = = ) < 
PrRrIops. = = a 5 
rE errDF DE = — 7" Yt 
NAMES Nn n n n n 
oS i eo 2 oO 
5) D rD) 0) o 
OF = Nn = mn = 2) = mM rd n 
See S25 8285 g25 828 
ORDERS. © oF D5 ® A Sok 
Dep Me Dep Mma) me DY 
Peete etity. fe ci 4G Oo woh) 456 8 4 726 


By ELEMENTARY ARITHMETIC. 


ORDERS OF FIGURES LOWER THAN UNITS. 


22. As orders of figures represent values of units, and 
values higher than units, so may they represent values 
lower than units, as 

In all orders of figures a figure written in one order rep- 
resents a value one tenth of the value represented by the 
same figure one order to the left; so a figure written next 
to the right of units represents a value one tenth of the 
value represented by the same figure in the order of units. 
Tenths of units are tenths. Therefore, the order occupied 
by the figures written next to the right of units is called 
the order of tenths, and is always separated from the order 
of units by a mark called the Decimal Point. 

23. The order occupied by the figure written next to 
the right of tenths is called hundredths, and the order occu- 
pied by the figure written next to the right of hundredths 
is called thousandths. 'These three orders of tenths, hun- 
dredths, and thousandths, form a group of orders called the 
Period of Thousandths. 

Thus, 245.364 is a number composed of 245 wnzis and 364 thou- 
sandths, or of two periods, a period of units and a period of thou- 
sandths: and is read two hundred forty-five and three hundred sixty- 
four thousandths. 

24, It will be observed that the principles governing 
the order of units and those that are higher, have equal 
application to the orders lower than units; that is, they are 
written in the scale of ten. Those orders lower than units 
are called Decimals, and all operations performed upon 
them are identical with corresponding operations upon 
integral numbers. 

25. United States Money is written in accord- 
ance with this feature of notation. The order of units and 
those that are higher are represented by dollars, the tenths 
and hundredths by cents, and the thousandths by mills. 

Thus, $3.375 is read three dollars, thirty-seven cents and five mills. © 


$5.098 is read five dollars, nine cents and eight mills; and 
$8.406 is read eight dollars, forty cents and six mills. 


NOTATION AND NUMERATION. 33 


26. The character $ means dollars, and is written at 
the left of the number, and is supposed to be a monogram 
of the letters U.S. 

The cents and mills may be read as a decimal of the dol- 
lar, or the mills as decimal of the cent. 

Thus, $6.236 may be read s7x and two hundred thirty six thousandths 
dollars, or it may be read s¢x dollars, twenty-three and six-tenths cents. 

27. Dercimat Notation anp UNITED States Money 
COMPARED. 


Decimal Notation. U. S. Money. 

56.20 is read fifty-six and $56.20 is read fifty-six 
two tenths, or twenty hun- | dollars and twenty cents. 
dredths. 


63.45 is read sixty - three $63.45 is read sixty - three 
and forty -five hundredths. dollars and forty-five. cents. 

79.306 is read seventy - five $75.356 is read seventy-five 
and three hundred fifty-six | dollars, thirty-five cents and 
thousandths. six mills. 


SLATE EXERCISES. 

Write in figures: 

1. Seven. 

2. Twelve. 
3. Seventeen. 
4. Twenty. 
5. Thirty - five. 
6. Fifty-nine. 
7. Ninety-seven. 
S. Two hundred nineteen. 
9. Three hundred eighty - six. 
IO. Six hundred three. 
11, Nine hundred forty. 
12. One thousand, six hundred fifty - four. 
13. Four thousand, nine hundred sixteen. 


14. Seven thousand, seventy - nine. 
A2* 


34 ELEMENTARY ARITHMETIC. 


15. Nine thousand, eight. 

16. Twelve thousand, nineteen. 

17. Sixteen thousand, four hundred seventy - six. 

18. Thirty-five thousand, two hundred eleven. 

19, Fifty-four thousand, three hundred ninety - seven. 

20. Sixty -six thousand, six hundred sixty - six. 

21. Ninety-four thousand, eighteen. 

22. One hundred twenty-five thousand, two hundred 
sixteen. 

23. Two hundred eleven thousand, nine hundred eleven, 

24. Six hundred three thousand, eight hundred eight. 

25. Nine hundred seventy - five thousand, sixty. 

26. One million, three hundred twenty-six thousand, 
four hundred seventy - nine. 

27. Two million, seven hundred thousand, five hundred 
ninety - eight. 

28. Three million, sixty-four thousand, two hundred 
eighty - one. 

29. Four million, four thousand, four hundred. 

30. Five million, five thousand, five. 

31, Seventy-two million, nine hundred forty - five thou- 
sand, eight hundred sixteen. 

52. Forty-seven million, fifty-nine thousand, six hun- 
dred fifty - four. 

33. Sixty million, sixty thousand, six hundred. 

34. Eighty million, eighty thousand, eighty. 

35. Thirty-nine million, eight thousand, nine. 

36. Fifty-six million, three hundred twenty thousand, 
nine hundred sixteen. 

37. Twenty-nine million, one hundred seven thousand, 
ninety - four. 

38, Seventy-five million, twenty-eight thousand, nine. 

39. Two hundred forty-seven million, six hundred twea- 
ty - five thousand, eight hundred sixteen. 

40. Five hundred forty million, three hundred eighty- 
seven thousand, twenty - one. 


NE 


NOTATION AND NUMERATION. 


30 


41, Six hundred three million, eighty-nine thousand, 


seven. 


42. Seven hundred fifty-six million, eight thousand, 


forty. 


43. Hight hundred million, eight hundred thousand, 


eight hundred. 


44. Nine hundred ninety-nine million, nine thousand, 


nine, 


Write in words, read from the page, or write in figures from dicta- 


tion. 


SO 20 S32 Se Go iS N 


17 
26 
38 
43 
d1 
67 
76 
84 
95 
100 
119 
236 
328 
456 
508 
643 
7d9 
804 
973 
1000 
1325 
2140 
d254 
4562 
5674 
63507 
7040 


8036 
9003 
10600 
10128 
253564 


37428 


45546 
52360 
64208 
73064 
80975 
93602 
100000 
123250 
267580 
329705 
463076 
520362 
605729 
750038 
800201 
900037 
1000000 
1234526 
2362340 
3798709 
4569044 


5120674 
6204638 
7035690 
8002637 
9200075 
LOOO0000 
18124620 
23742635 
30428640 
463570904 
57298027 
693803824 
72506241 
83037256 
90364562 
LOOOO0000 
123234345 
231456213 
3259368450 
432523609 
561431035 
632520362 
175306728 
821078584 
950364260 
LOOQ000000 
9909909909 


36 HLEMENTARY ARITHMETIC. 


REVIEW QUESTIONS. 


What is a Unit? Give example. What does one denote? How 
find the several successive numbers from one? What is a Number ? 
Give example. What is Arithmetic? Denominate Number? Abstract 
Number? Give examples of each. What is the Unit of a Number? 
Illustrate. What are Like Numbers? Unlike Numbers? Give ex- 
amples of both. When are Units of the same kind? Examples. When 
are Units of different kinds? Examples. What are Simple Numbers ? 
Give examples. To which kind of numbers do Simple Numbers 
belong? What are Compound Numbers? To which kind of num- 
bers do Compound Numbers belong? What is an Integer? Example. 
What is a figure? 

What is Notation? How many methods of Numerical Notation ? 
What kind of characters are used by each? Name the characters 
used in the Roman Notation, and their values. What is the first 
principle that governs the combinations of these letters to express 
number? Second principle? Third principle? Name the charac- 
ters employed in the Arabic Notation. What is the use of the 
Cipher? What are the other nine figures called? What do they 
severally represent? How are Orders of figures represented? Tllus- 
trate by an example. What is the first principle governing the com- 
bination of figures to express number? The second principle? The 
third principle? The fourth principle? Name the first ten of these 
orders. How many orders are used to form one period? Name the 
first five periods. How is absolute value represented? Local value? 
Give examples of each. 

What is Numeration? Give rule for Notation. Rule for Numer- 
ation. Name the order of figures next to the right of units. What is 
the second order below units called? The third order? What group 
do these three orders form? What separates the order of tenths from 
the order of units? What name is given to the orders lower than 
units? In what scale of numbers are they written? What are 
the names used to express United States Money? Which of these 
correspond to the order of units and those orders that are higher ’ 
To tenths and hundredths? To thousandths? What does the charac- 
ter $ signify? How may cents and milis be read? Mills ? 


ADDITION. 37 


SH 4G JEILO px pad Ol 


ADDITION. 


Art, 28. In 6 there are six units, and in 4 there are 
four units. In 6 and 4, which are 10, there are six units and 
four units, equal to ten units. 10 therefore contains as 
many units as 6 and 4, and is called the swm of 6 and 4. 

10 is also the sum of 7 and 3, or 5 and 3 and 2, or of any 
two or more numbers that contain as many units. Hence, 

29. The Sam of two or more numbers is the number 
that contains all their units and no more. 

Thus, 9 is the sum of 4 and 5, or 6 and 3. 

30. Addition is the process of obtaining the sum 
of two or more numbers. 

31. The Sign of Addition is +, called plus. When 
placed between two numbers it indicates the addition of 
one number to the other. 


Thus, 5+-7 signifies that 7 is to be added to 5, or that 5 is to be 
added to 7. 


32, Addends are the numbers to be added. 

Thus, 5+4+3-+8=20. 5, 4, 3, and 8, are addends. 

33. The Sign of Equality is =, and is read is 
equal to, or equals, and when placed between two quanti- 


ties it signifies that the quantity on one side is equal to the 
quantity on the other side. . 


~ Thus, 5+-7=12; or, 44+5=38-+46; or, 8=3-L5. 


PRINCIPLE. 


Numbers can be added only to /ike numbers. 


38 HLEMENTARY ARITHMETIC. 


ORAL PROBLEMS. 


1. A boy paid 25c. for a First Reader, and 10c. for a 
slate. How much did he pay for both? 

SoLuTion.—Since he paid 25c. for a First Reader, and 10c. for a 
slate, he paid for both the sum of 25c. and 10c., which is 35c. 

2, A girl paid 30c. for a Second Reader, 12c. for a 
writing-book, and 8c. for a sponge. How much did she pay 
for all? 

3. 20c. was paid for a drawing-book, 15c. for a speller, 
and 9c. for a pencil. How much was paid for all? 

4. John has 18 marbles, James 12 marbles, and George 
9 marbles. How many have they all? 

&. Mary had 382c., her father gave her 11 more, and her 
mother 8 more. How many did she then have? 

6. How many bushels of apples will 3 men pick in one 
day, if the first pick 16 bushels, the second 12 bushels, and 
the third 11 bushels? 

7. Minnie, Emma, and Jane went out berrying; Minnie 
gathered 21 pints, Emma 15 pints, and Jane 12 pints. How 
many pints did they all gather? 

S. A gentleman has 12 books on one table, 10 on 
another, 9 on a third table, and 8 on the fourth. How 
many has he on the four tables? 

9. A man sold 10 turkeys to one man, 9 to another, 8 
to a third, and kept 7. How many turkeys had he at 
first? 

10. A gentleman kept 15 sheep in one field, 9 in another, 
8 in a third, and 7 in a fourth. How many sheep did he 
have? 

17. A woman paid $12 for a shawl, $10 for a bonnet, $8 
for a chain, and $6 for a pair of boots. How much money 
did she expend? 

12. A gentleman bought a sleigh for $45, a set of har- 
ness for $15, a robe for $7, and a whip for $5. How much 
did he pay for all? , 


ees hc CU 


—————— oe er 


ADDITION. 39 


13. I saw 36 birds in one flock, 14 in another, 12 ina 
third, and 8 in a fourth. How many birds did I see? 

14. A lady put up 30 quarts of cherries, 20 quarts of 
plums, 15 quarts of peaches, and 9 quarts of tomatoes. 
How many quarts of fruit did she put up? 

15, A man went ona journey of four days; the first day 
he traveled 28 miles, the second 12 miles, the third 15 miles, 
and the fourth 9 miles. How far did he travel? 

16. Bought 20 yards of muslin from one merchant, 20 
yards from another, and 30 yards from a third. How many 
yards did I buy? 

17. Gave $9 for a pair of boots, $8 for a coat, and 90c. 
for a pair of gloves. How much did they all cost? 

1S. A boy wrote 20 words in one minute, 16 the second, 
10 the third, 8 the fourth, and 6 the fifth, How many 
words did he write in five minutes? 

19. Gave $5 for a hat, 50c. fur a neck - tie, $4 for a vest, 
and 40c. for a pair of cuffs. How much was given for all? 

20. James earned $8, and William $7 in one week, and 
their father as much as both. How much did they all 
earn? 

21, Paid 20c. for 12 oranges, and 30c. for 15 oranges. 
How much money did I expend? How many oranges did 
I buy? 

22. Paid 50c. for 13 lemons, and 40c. for 9 lemons. How 
many lemons did I buy? How much did they cost? 

23. John worked 7 days for $4, and James 8 days for $7, 
and Rufus as many days as John and James for $11. How 
many days did they all work? How much did_ they 
receive? 

24, Sold a chain for $11, a watch for $9 more, and a cow 
for as much as both watch and chain. How much did I get 
for all? 

25, Spent- $5 Monday, $7 Tuesday, $9 Wednesday, $5 
Thursday, $2 Friday, and 50c. Saturday. How much did I 
spend during the week? 


40 HLEMENTARY ARITHMETIC. 


26. Paid $3 for 20 quarts of berries, $2 for 15 quarts, 
and 60c. for 5 quarts. How many quarts of berries did I 
buy? How much did they cost? 

27. Gave 30c. for 2 pounds of beef, 40c. for 3 pounds of 
pork, and 20c. for 3 pounds of mutton. How many pounds 
of meat did I buy? How much did I pay for meat? 

28, Aman bought a sleigh for $20, a harness for $15, 
and sold them at a gain of $9. How much did he receive 
for them? 

29, A cage was made of 12 wires on each side, 8 on 
each end, and 30 in the roof. How many wires were re- 
quired to make the cage? 

30. How many strokes does a clock make in 12 hours, 
that strikes the hours, and 1 each half hour? 

31. A room is 20 feet long, and 15 feet wide. How 
many feet long is a line that would reach entirely around 
the room? 

3&2. A gentleman upon a journey traveled for 6 days, in- 
creasing the distance traveled each day by 5 miles. If he 
traveled 10 miles the first day, how far did he travel during 
his journey? 

33. Paid $12 for books, $18 for clothes, and as much for 
a horse as for both, plus. $10. How much money did I 
expend? 

34. A person walked 20 miles from home in one day, 
and the second day 30 miles farther, the third day he rode 
the entire distance home again. How far did he travel 
during the three days? 

30. Two houses are 6 miles apart. How far will that 
person travel who starts from one of them, visits the other, 
and returns 5 times? 

36. In an orchard there are 12 cherry trees, 16 pear 
trees, and four more apple trees than pear trees. How many 
trees are there in the orchard? 

37. A forester cut 15 cords of hickory wood, 20 cords of 
beech wood, 15 cords of oak wood, and 10 cords more of 


—————— CO 


ADDITION. | 41 


maple wood than he cut of beech wood. How many cords 
of wood did he cut? 

38. Aman bought one lot of flour for $30, and another 
lot for $35. He sold the first lot at a gain of $10, and the 
second lot at a gain of $15. How much did he receive for 
both lots of flour? 

39, had 3 flocks of geese, the first containing 28, the 
second 22, and the third 25. I subsequently added 10 to 
each flock. How many geese did I then have? 

40. A boy who had 4 bags, put into each one 15 nuts at 
one time, 10 into each at another time, and the third time 
he put 12 into each of three of them, and 14 into the fourth 
one. How many nuts are in the bags? 

41. Bought 6 barrels of flour at $12 a barrel, and sold 
them at a gain of $9. How much did I receive for them? 


ANALYSIS OF ADDITION. 


42. What is the sum of 35645434879? 


PROCESS. Anatysis.—l. Write the addends so that figures of 
- the same order shall stand in the same column. 

356 2. Add the figures in the column of units; thus, 8, 

543 12, 18 units, equal to 1 ten and 8 units. Write the 8 units 


79 in the order of units, and add the 1 ten to the column of 
tens; thus, 
1778 3. 1,8, 12, 17 tens, equal to 1 hundred and 7 tens. 


Write the 7 tens in the order of tens, and add the 1 hun- 
dred to the column of hundreds; thus, 

4. 1, 9, 14, 17 hundreds, equal to 1 thousand and 7 hundreds. 
Write the 7 hundreds in the order of hundreds, and the 1 thousand in 
the order of thousands. 

The sum of 356-+-543--879 is therefore 1778. 


NOTE 1. Pupils should not be permitted to acquire the habit of saying 9 and 3 
are 12 and 6 are 18, ete. Itis better to name results only, in each addition. 

2. It will give greater facility in operations of addition, and lead to the detec- 
tion of errors, to require pupils to add in a reverse order. 


42 ELEMENTARY ARITHMETIC. 


WRITTEN EXERCISES. 


43. Add 9 thousand 3, 58 thousand 68, 64 thousand 208, 
99 thousand 99, 273 thousand 574, 540 thousand 730, 879 
thousand 7, 386. 

44, Add 12 thousand 29, 79 thousand 524, 357 thousand 
58, 792 thousand 218, 854 thousand 679, 927 thousand 9, 
678 thousand 75, 2368. 

4. Add 239 thousand 316, 528 thousand 97, 2 million 
365 thousand 297, 7 million 67 thousand 954, 9 million 
95 thousand 658, 8 million 586 thousand, 9 million 
209. 

46. Add 538 thousand 297, 3 million 3,5 million 5 thou- 
sand 555, 6 million 72546, 9 million 75 thousand 654, 10 
million 10 thousand 10, 12 million 807307. 

7. Add 5 million 467 thousand 956, 729 thousand 475, 
3 million 250 thousand 563, 17 million 368 thousand 845, 
1729, 12 thousand 699, 4 million 37 thousand 76, 36 million 
754 thousand 684, 9 million 500. 

48. Add 236 million 827 thousand 973, 56 million 638, 
829 thousand 976, 5 million 38 thousand 647, 798 thousand 
364, 45 million 7, 7 thousand 8, 375 million 984 thousand 
879, 99 million 99 thousand 99. 

49, Add 3 thousand 709, 53 thousand 678, 4 million 379 
thousand 863, 7 million 620 thousand 308, 495 thousand 678, 
7 million 875 thousand 689, 29844. 

00, Add 47854, 328 thousand 567, 2 million 47 thousand 
854, 579 thousand 863, 4 million 876 thousand 594, 3 million 
2 thousand 8, 97 thousand 582, 9010987. 

51. Add 87 million 964 thousand 757, 3 million 986 
thousand 759, 975 thousand 368, 77 thousand 675, 7 thou- 
sand 854, 989, 78, 9, 123674, 8478 

52. Add 43 thousand 687, 56 thousand 437, 64 thousand 
482, 75 thousand 438, 79 thousand 347, 86 thousand 596, 27% 
thousand 368, 39 thousand 564. 

58. Add 728 thousand 756, 689 thousand 476, 8 million 


ADDITION. 43 


674 thousand 758, 12 million 846 thousand 378, 56 thousand 
478, 9 thousand 64, 462 thousand 965, 778899. 

54. Add 678 thousand 345, 756 thousand 987, 545 thou- 
sand 876, 787 thousand 456, 476 thousand 974, 856 thousand 
358, 798 thousand 567, 534 thousand 765, 739 thousand 748, 
593 thousand 584, 634 thousand 675, 587 thousand 378, 798 
thousand 786. 

55. Add 543 thousand 678, 789 thousand 657, 678 thou- 
sand 545, 654 thousand 787, 479 thousand 674, 853 thousand 
658, 765 thousand 897, 567 thousand 435, 847 thousand 9387, 
485 thousand 395, 576 thousand 436, 783 thousand 7895, 687 
thousand 897. 

56. Add 4 million 534 thousand 897, 3 million 778 thou- 
sand 697, 5 million 769 thousand 758, 6 million 836 thousand 
745, 757 thousand 894, 367 thousand 376, 675 thousand 487, 
456 thousand 737, 568 thousand 976, 79 thousand 754, 65 
thousand 438, 79 thousand 768, 68 thousand 479, 82 thou- 
sand 397, 8 million 8 thousand 8. 

57. Add 276 million 456 thousand 787, 956 million 876 
thousand 545, 679 million 877 thousand 453, 466 million 788 
thousand 567, 575 million 867 thousand 238, 698 million 458 
thousand 487, 756 million 789 thousand 789, 469 million 756 
thousand 574, 574 million 567 thousand 978, 885 million 745 
thousand 765, 769 million 354 thousand 657, 433 million 778 
thousand 844, 667 million 332 thousand 266, 589 million 746 
thousand 457, 521 million 564 thousand 653. 

5S. Add 787 million 654 thousand 672, 545 million 678 
thousand 659, 354 million 778 thousand 664, 832 million 768 
thousand 575, 754 million 854 thousand 896, 987 million 987 
thousand 657, 475 million 657 thousand 964, 765 million 887 
thousand 976, 879 million 765 thousand 475, 567 million 547 
thousand 588, 756 million 453 thousand 967, 448 million 877 
thousand 334, 662 million 235 thousand 776, 754 million 647 
. thousand 985, 356 million 463 thousand 125. 


44 


NOTE.—For examples 59 to 83 inclusive, read across the page. 


HLEMENTARY ARITHMETIC. 


els 


4689 
7568 
D674 
6706 
7068 
8746 
D978 


8364 
D799 
8658 
67384 
7a09 
6874 
7049 
6793 


dd47 
8796 
5647 
9876 
1837 
8768 
5678 
1847 
8789 
7648 


679 
543 
689 
798 


104. 
15467 
54874 
435675 
89763 
34576 
87658 
48326 


75897 
89784 
76348 
57867 
76845 
54597 
68745 
85697 


‘58746 


76568 
67875 
98362 
95764 
69897 
19645 
65459 
79876 
86489 


105. 


543876 
706836 
647479 

16045 
786456 
4AV5679 
187536 


569478 
678797 
956746 
758667 
876748 
456685 
874896 
568437 


875968 
968475 
697989 
823424 
796848 
863685 
697876 
168706 
897678 
976578 


L106. 


7983648 
5867437 
4976877 
7387586 
8975438 
5467679 
4857748 


8485675 
5676487 
7834656 
5847589 
8795654 
1047878 
8567057 
5784995 


8659448 
7586876 
8979789 
9106758 
T647576 
5638975 
8756687 
6574835 
7689978 
4898767 


LO F: 

6785468 
7584786 | of 
8376854 | A 
9367973 } Qn’ 
4756856 | 96 
7678457 | 99 
4856738, 
T948567 
8769858 | 90. 
7989998 | 91. 
8888888 | 92. 
VY57BY ( 93. 
9696699 | 94. 
7685785 | 95, 
8659897 
5784546 ) 
8638756 
7987899 | 96. 
8956785 | 97. 
8678598 | 98. 
3738957 f 99. 
9763687 | 100. 
8765868 | 101. 
9776898 
S987976 | 


For examples 84 to 89 inclusive, read vertically the successive columns 
within the limit of the brace;and so for examples 90 to 95 inclusive, 


and same manner for exampies 96 to 101 inclusive. 


For examples 102 to 107 inclusive, read vertically the entire column. 


108. Add 
29 dollars 59 cents 6 mills. 
48 dollars 27 cents 4 mills. 
87 dollars 75 cents 7 mills. 
98 dollars 87 cents 5 mills. 

127 dollars 7 cents. 

478 dollars 9 cents 6 mills. 


PROCEsS. 
$29.596 
48.274 
87.757 
98.875 
127.07 
478.096 


$869.668 


ADDITION. 45 


PEO SLL ELL 119. 120. 


109. $5.607 | $36.089 | $59.006 | $68.02 |$85.308 | $165. 
110. 1.81 | 45.004! 67.507| 74.326! 97.009! 276.06 
111. 12.563 | 38.25 | 59.06 | 86.005 /128.03 | 359.007 


ti be ae 978 0d2| 3. Ot? 43998.909 
113. 139.875 | 48.764] 57.872] 49.078 1584.27 | 875. 
114, 237. 9.006 | 8.957 06 | 263. 896.08 


WRITTEN PROBLEMS. 


121. Paid $367 for a horse, $2698 for a house, and $479 
for furniture. How much did I pay for all? 

122. A lot cost me $3964, a barn $996, fencing $1387, 
and farming tools $178. What did they all cost? 

125. A ship sailed in one week 846 miles, the second 
week 958 miles, and the third week as much as the other 
two. How far did she sail in the three weeks? 

124. Three vessels are loaded with lumber, as follows: 
the first carries 29368 feet, the second 1986 feet more than 
the first, and third 7568. feet more than the second. How 
many feet do they all carry? 

125. Bought 4 car loads of wheat, paying for the first 
$878.36, for the second $1179.756, for third $1288.08, and 
for the fourth $1587.128. How much did I pay for the 
wheat? 

126. Built a block of four houses; the two end ones cost 
me $4897.603 each, and the two middle ones $3579.875 
each. What did the block cost me? 

127. A and B each paid $29768.30 taxes, and C and D 
each $18679.796. How much did they all pay? 

128. The water tax receipts during the first week of 
February,1875, were $9367.50, the second week $8576.86, 
the third. week $11975, and the fourth week $9987.87. 
What were the receipts for February? 

129. Bought a house for $5897.48, a lot for $3673.62, 
and sold them both so as to gain $1976.50. How much did 
I receive for them? 


46 ELEMENTARY ARITHMETIC. 


130. Stocked four farms with sheep, putting into the 
first 5467, the second 4264, the third 876 more than the first, 
and the fourth 975 more than the second. How many sheep 
did I have? 

1381. A,B andC build a railroad, A receiving $12873.64 
for his share, B $13560.92, and C as much as A and B and 
$589.78 more. How much did the railroad cost? 

132. Areal estate dealer sold 2 lots for $2976.40 each, 


and 2 others for $2745.80 each. How much did he receive 


for them all? 

138. Bought 256 bushels of wheat for $378.75, 267 
bushels of oats for $198.68, 176 bushels of corn for $119.96, 
and 398 bushels of hay seed for $1367.75. How much did 
they all cost? How many bushels were bought? 

134. Bought a house for $4688, a lot for $3650; I sold 
the house at a gain of $627, and the lot at a gain of $565. 
How much did I receive for both? 

135, A farmer raised in one year 698 bushels of corn, 
the second 987 bushels, the third. 1289 bushels, the fourth 
1565 bushels, and the fifth 1976 bushels. How much did 
he raise in the 5 years? 

136. A flouring mill turned out in one week 1089 barrels 
of flour, in the second week 999 barrels, in the third 1467 
barrels, and in the fourth 264 barrels more than in the 
second week. How many barrels were turned out in the 4 
weeks? 

137. Your drovers sent sheep to market, the first send- 
ing 976, the second 1079, the third 1285, and the fourth as 


many as the first and second. How many did they all | 


send? 

138. A,B,C and D, are farmers; A made in one year 
$1365, B $376 more than A, C $256 more than B, and D 
$468 more than C. How much did D make? How much 
did they all make? 

139. A gentleman traveled in January 1687 miles, in 


February 987 miles, in March 1279 miles, and in April 279 


ADDITION. 47 


miles farther than in February. How far did he travel in 
the 4 months? 

140. A girl can count 5789 in one hour. How many 
can she count in 6 hours? 

141, At a saw-mill there were cut in one week 16728 
feet of lumber. How many feet can be cut in four weeks? 
142. If one car contains 659 bushels of wheat, how 
many bushels in a train of 7 such cars? 

45, Paid $5674.25 for one house. How much should I 
pay for 5 such houses? 

144, Paid $3560 for a house, and $1880 for a lot. How 
much must I pay for 3 such houses, and 2 such lots? 

145. A certain elevator will hold 6759 bushels of wheat, 
and 4678 bushels of corn. How many bushels of wheat 
and corn will 4 such elevators hold? 

146. How many bricks may a mason put into 6 walls, 
if he can put 6798 bricks in one wall? 

147. A railroad company carries 5375 passengers in one 
week. At the same rate how many passengers can they 
carry in 8 weeks? , 

148. A company earns $7987.60 during the month of 
January. At the same rate how much will they earn in the 
first four months-of the year? 

149. There are 5760 grains in one pound of silver. 
How many grains are there in 6 pounds of silver? 

150. Té it cost $12326 to build one mile of railroad, how 
much will it cost to build 6 miles? 

151. If sound moves 1142 feet in one second of time, 
how far will it move in 10 seconds? 

152. Tf acarrier pigeon can fly 987 miles in one day, 
how far can it fly in one week? 

153. An army was composed of 9 regiments, and there 
were 976 men in each regiment. How many men in the 
army ? 

154. There are 365 days in one year. How many days 


in 20 years? 


48 ELEMENTARY ARITHMETIC. 


Se ANT vee 


SUBTRACTION. 


Art, 34, Since 6 units and 4 units are 10 units, 10 
units are 4+ units greater than 6 units, or 6 units are 4 units 
less than 10 units; also, 

Since 12 units are 4 units more than 8 units, or 8 units 
are 4 units less than 12 units, 4 units is the difference of 
any two numbers one of which contains 4 units more or less 
than the other, hence 

35, The Difference of two numbers is the number 
of units which one of the numbers is greater or less than 
the other. 

Thus, 4 is the difference of 9 and 5. 


36. Subtraction is the process of obtaining the 
difference of two numbers. 

3%. The terms employed in subtraction are minuend, 
subtrahend and difference. 

38. The Minaend. is the larger of the two numbers 
whose difference is required, and is the number to be ° 
diminished by the operation of subtraction. 

Thus, 7—3=4, 7 is the minuend. 

39, The Subtrahend is the less of the two numbers 
whose difference is required, and is the number to be sub- 
tracted from the minuend. 

Thus, 7—3=—4, 3 is the subtrahend. 

40, The Sign of Subtraction is —, called minus. 
When placed between two numbers it denotes that the 


SUBTRACTION. 49 


number on the right of the sign is to be subtracted from 
the one on the left. 

Thus, 12—9=8. 

Nore.—The difference is sometimes called the remainder, especially when a part 


of a number is to be subtracted from the whole number; as, 5 yards of muslin are 
cut from a piece containing 20 yards. : 


PRINCIPLE. 


The minuend and subtrahend are always like numbers. 


ORAL PROBLEMS INVOLVING BUT ONE PROCESS. 


1. A boy had 20c., and paid 1dc. for a slate. How 
much had he left? 
ANALysts.—Since he had 20c., and paid 15c. for a slate, he had left 


as many cents as 20c. is greater than 15c., which are 5c. Therefore 
he had dc. left. 


2. Paid 14c. for paper, and 9c. for a pencil. How 
much more was paid for the paper than for the pencil? 

3. Mary is 16 years old, and her brother is 8 years old. 
What is the difference of their ages? 

4. John bought 18 marbles, and James 12. How many 
more did John buy than James? 

}. A lady having $19, paid $8 for a shawl. How much 
did she have left? 

6. A boy gathered 21 quarts of nuts, and sold 12 quarts. 
How many quarts were left? 

7. A man bought 28c. worth of meat, and gave the 
trader 35c. How much change should he receive? 

S. Bought 38c. worth of cotton cloth, and gave the 
merchant 50c.. How much change should I receive? 

9. Bought a suit of clothes for $36, and gave the dealer 
$50. How much change should I receive? 

10. There are two trees in my yard, one of which is 48 

feet high, and the other is 13 feet less. How high is the 
other tree? 


50 ELEMENTARY ARITHMETIC. 


11, A gentleman is 60 years old, and his wife is 12 years 
younger. How old is the wife? 

12. A farmer raised 75 bushels of corn this year, which 
is 15 bushels more than he raised last year. How much did 
he raise last year? 

13, Paid $90 for a wagon, and sold it so that I lost $20. 
How much did I receive for the wagon? 

14. Took 80 bushels of oats to market, which is 30 
bushels more than I left at home. How many bushels did 
I leave at home? 

15. Sold 100 bushels of wheat, which was 30 bushels 
more than I kept. How many bushels did I keep? 

16. Bought 88c. worth of groceries, and gave the grocer 
$1. How much change should I receive? 


ORAL PROBLEMS INVOLVING TWO PROCESSES. 


17. Paid 7c. for a pencil, 12c. for a book, and gave 5c. 
to the dealer. How much change should I receive? 

18. A lady had $12, found $10, and afterward lost $8. 
How much did she have left? 

19, Had 50c., paid 30c. for a book, and 12c. for toys. 
How much had I left? 

20, A farmer kept 45 hogs in 3 pens; in one pen were 
20 hogs, in the second 15. How many were there in the 
third? 

21. Mary had 12 peaches, Jane 9 peaches, and Sarah 30 
peaches. How many peaches has Sarah more than both 
Mary and Jane? 

22. A wagon cost $50, a sleigh $20, and harness $10. 
How much does the wagon cost more than both sleigh and 
harness? 

23. Bought 40c. worth of sugar, and 30c. worth of 
coffee. How much change should I receive if I gave the 
grocer $1. 


24. A hogshead contains 63 gallons of water. If I pour 


SUBTRACTION. 51 


into it at one time 40 gallons, at another 14 gallons, how 
many gallons more will fill it? 

25. A tree is 75 feet high. If 40 feet be broken off at 
one time, 20 feet at another, how much will yet remain? 

26. A boy put 18 chickens into one coop, 12 into 
another, and enough in the third to make 45 altogether. 
How many did he put into the third coop? 

27. A woman paid 50c. for some tea, 25c. for cheese, 
and enough for oil so that they all cost $1. How much did 
she pay for oil? 

28. <A farmer planted an orchard of 100 apple trees in 4 
days. ‘The first day he planted 30 trees, the second 30 trees, 
the third 20 trees. How many did he plant the fourth day? 

29. A hunter in 4 days shot 80 birds. The first day he 
shot 24 birds, the second 24 birds, and the third 12 birds. 
How many did he shoot the fourth day? 

30. A gentleman lost $20 each day for 3 days, and 
enough the fourth day to make his whole loss $80. How 
much did he lose on the fourth day? 

31. From a pile of wheat containing 72 bushels, 20 
bushels were taken at one time, 30 bushels at another, and 
10 bushels at another. How many bushels were left? 

32, Gave 16c. for a pound of sugar, 20c. for a pound of 
coffee, and 45c. for meat. How much more was given for 
meat than for both sugar and coffee? 

33. Gave 20c. for oranges, 30c. for lemons, and 41c. for 
candies. How much less did I pay for candies than for the 
lemons and oranges? 

34. Sold a pencil for 9c., a slate for 12c., and some toys 
for 8c. If I received a 25c. piece, and a 10c. piece, how 
much change should I give? 

35. Bought a coat for $25, a pair of pants for $10, and 
a vest for $6. If I give a $50 bill, how much should I re- 
ceive in change? 

36. A lady put up at one time 22 quarts of cherries, at 
another 10 quarts, and at another 8 quarts. She lost at one 


Sy ELEMENTARY ARITHMETIC. 


time 12 quarts, at another 9 quarts. How many quarts 
were left? 

37. A boy had 9 marbles, found 3 more than he had. 
He afterward lost 6 less than he found. How many marbles 
did he have left? 

38. A squirrel ran up a tree 20 feet, then down 8 feet, 
up 4 feet, down 12 feet, up 9 feet, up 7 feet, down 11 feet, 
up 7 feet, down 6 feet, down 9 feet, up 25 feet, down 10 
feet, down 9 feet, up 12 feet, down 16 feet. How high is 


the squirrel ? 
ORAL COMBINATIONS. 


39, 184+74+8+9—7—6—448—54+9—7+464+9=? 
9012-29-75 -19 10-62-14 4 bho 
41. %7+8+6—9—34+6+411—10—8+45+44—6—5=? 
42. d0—8—7—6—54+7—544—548—9+46—8412=? 
48. 184+12—8—8—8+4124+544—-10—7+4+5+4+3—9=? 
44. 164+8—9+15—9—9+6—9+45—741245—20=? 
45. 244+8+8—20+749—8+6—449+6—5—10=? 
46. 94+6—84+7+4+8+4+8—6—6412410+410—20=? 
47. 124+5+5+5—7—746—54+94+9—8—648+46=? 
48, 18—6—4—8+11+5—10+4+2048+4+7+49—40=? 
49, 124+54+4+4+38—5—54948+49+4+8—5—5—5—5=? 
60. %7+8+9—3+9—349—3+9—349+4+3-—9-—2=? 
61, 174+-7—124746—54+9—4484+7—84646=? 
62. 164+12+12—10—104+8—9+4+7—94+6—9412=? 
63. 8+94+6+6—11—9412+5+4—-10—9—8+6=? 
b4. 114+9—6411—5 +204+20-— 504214+9—8—12=? 
55. 60—4—4—4—4—4—4-— 4-4-4484 8-—7_-7=? 
56. 40+40+20+60—100—49+80—40—50+90=? 
b7. 124124124+12412412—2—10—20—30+90=? 
HS. 11-1 1d $11 11 6 eee 
59s -25+-10+-53—8— 7 +11 —12-4.6—20 419-6 =e 
60. 30+20—40+15+25+450+100+4200+4400+800=? 
61. 500—300—100—50+4-25+-25—30—30—30—10=? 
62. 400—200—200-4+80+20+100+1000—200—500=? 
63. 1000—400—300—200+300+400+200—1000=? 


SUBTRACTION. 53. 


64. A boy had 144 chestnuts. He then took 7 away 
each day until there was left a number less than five. How 
inany were left? 

65. If there are 133 apples in a barrel, and George 
should take out 8 each day, and James 9 every other day, 
how many apples will George get if he precedes James? 

66. A man has 63 volumes of books in his library. 2 
shelves contain 9 yolumes each of poetry; 1 shelf Dickens’ 
works in 13 volumes, and Hood’s works in 5 volumes; an- 
other shelf Barnes’ Notes in 11 volumes; the remainder 
consists of 3 Bibles and the “ Little Classics.” How many 
volumes of ‘ Little Classics” in the library? 


ANALYSIS OF SUBTRACTION. 


67. From 9876 subtract 3452. 


ANALYsis.—l. Write the numbers so that figures of 


PROCESS. the same order shall stand in the same column. 
9876 2. Begin at the order of units thus: 6 units less 2 
3452 units are 4 units. Write the 4 units beneath the column 
of units. 


6424 3. 7 tens less 5 tens are 2 tens. Write the 2 tens be- 
neath the column of tens. 
4. 8 hundreds less 4 hundreds are 4 hundreds. Write the 4 hun- 
dreds beneath the column of hundreds. 
5. 9 thousands less 3 thousands are 6 thousands. Write the 6 
thousands beneath the column of thousands. 
Therefore, 9876—3452—=6424. 


6S. From 8345 suptract 4879. 


ANALYsIs.—1. Write the numbers as above. 


PROCESS. — 9._-»5 units of the minuendare less than 9 units of the 
8545 — subtrahend. To supply this deficiency, take 1 ten of the 
4879 4 tens of the minuend, and change it to units. 1 ten is 


ae. equal to 10 units, which, added to the 5 units is equal to 

3466 15 units. 15 units less 9 units are 6 units. Write the 6 
units in the order of units. 

3. 1 ten having been taken from the 4 tens of the minuend, 3 tens 

are left. 3 tens of the minuend is less than 7 tens of the subtrahend. 


54 ELEMENTARY ARITHMETIC. 


To supply the deficiency, take 1 hundred of the 3 hundreds of the 
minuend and change it to tens. 1 hundred is equal to 10 tens, which, 
added to the 3 tens is equal to 18 tens. 13 tens less 7 tens are 6 tens. 
Write the 6 tens in the order of tens. 

4. 1 hundred having been taken from the 3 hundreds of the minu- 
end, 2 hundreds are left. 2 hundreds of the minuend is less than 8 
hundreds of the subtrahend. To supply the deficiency, take 1 thou- 
sand of the 8 thousands of the minuend, and change it to hundreds. 
1 thousand is equal to 10 hundreds, which, added to 2 hundreds, is 
equal to 12 hundreds. 12 hundreds less 8 hundreds are 4 hundreds. 
Write the 4 hundreds in the order of hundreds. 

5. 1 thousand having been taken from the 8 thousands of the 
minuend, 7 thousands are left. 7 thousands less 4 thousands are 3 
thousands. Write the 3 thousands in the order of thousands. 

Therefore, 8345 —4879= 3466. 


69. From 2000 subtract 764. 


PROCESS. ANALYsIS.—1. Write the numbers as above. 
2000 2. 2000=1 thousand 9 hundreds+9 tens+1 ten. 1 
G4 ten=10 units. 10 units less 4 units are 6 units. 9 tens 
less 6 tens are 3 tens. 9 hundreds less 7 hundreds are 2 

1236 hundreds. 1 thousand less zero is 1 thousand. 


Therefore, 2000 —764= 1236. 


PROOF OF SUBTRACTION. 


The Subtrahend plus the Difference is equal to the 
Minuend. 


NoTeE.—The following examples will serve to illustrate the principles of sub- 
traction: 


1. ‘The minuend is 8469; the subtrahend is 7883. What 
is the difference? . 

2. The subtrahend is 7546; the difference is 2786. What 
is the minuend? 

3. The minuend is 15342 acres; the difference is 5679 
acres. What is the subtrahend? 

4. The subtrahend is $9457; the difference is $7638. 
What is the minuend? 
_  o. The difference is 12758 feet; the subtrahend 25697 

feet. What is the minuend? 


SUBTRACTION. 


WRITTEN EXERCISES. 


55 


NoTe.—The following table may be used in the following manner, for instance: 


Find difference of ex. 10, 2d col., and ex. 9, 3d col. 


Ix. 10, 2d col., is 9067, and ex. 9, 3d col., 65048. 


Ist col. 2d col. 
1.—567 3058 
2.—485 2360 
3.—369 4789 
4.484 3076 
5.—673 4508 
6.—767 5804 
71.—981 8054 
8.—867 6079 
9.—753 7906 

10.—862 | 9067 
11.—374 5432 
12.—219 4352 
13.—472 | 24385 
14.—865 7858 
15.—974 | 5897 
16.—749 | 9785 
17.—497 | 7589 
18.—673 5647 
19.—763 | 6475 
20.—850 | 4576 


3d col. 


7004 
74508 
45087 
60379 
73097 
84650 
46085 
58064 
65048 
79006 
60079 
90607 
58636 
83068 
63865 
97091 
71984 
49178 
87987 
27964 


4th col. 


567438 
875643 
706798 
197853 
368974 
486743 
675858 
864307 
705498 
890649 
347681 
475683 
643895 
436998 
698543 
756784 
465778 
875647 
587476 
835658 


5th col. 6th col. 

7643878 5683876 
5678545 6838765 
1754856 8378659 
7306871 3786598 
6785713 6597354 
4743658 9738465 
3657484 7384659 
9764583 3847396 
7489653 4659738 
4957862 8465973 
8956748 6000005 
7862495 7600005 
9578624 9856703 
5786249 5607398 
8624957 6709396 
5648947 9385607 
7564894 8000003 
4756489 5993601 
8946754 9000075 
6489475 8000004. 


WRITTEN PROBLEMS INVOLVING BUT ONE PROCESS. 


70. Paid $5431 for a house, and $2895 for a lot. 


much did the house cost more than the lot? 


PROCESS. 


$5431 
2895 


$2536 


How 


ANALYsis.—Since the house cost $5481, and the lot 


$2895, the house cost as many dollars more than the lot 
as $5431 is greater than $2895, which is $2536. 


71. Bought a cargo of wheat for $4526, and a cargo 
of corn for $2768. How much did the corn cost less than 


the wheat? 


56 HLEMENTARY ARITHMETIC. 


PROCESS. 
$4526 ANALYsIs.—Since the wheat cost $4526, and the corn 
268 $2768, the corn cost as many dollars less than the wheat 


as $2768 is less than $4526, which is $1758. 


$1758 
Norte. — The teacher will carefully impress upon the minds of the pupils the 
distinction of these Analyses. 


72. A mason put into the east wall of a building 1234 
bricks, and into the south wall 987 bricks. How many 
more bricks in the east wall than in the south wall? 

73. The population of a certain town in 1870 was 3467; 
and in 1874, 2399. How much less was the population in 
1874 than in 1870? 

74. A gentleman’s income one year was $2958, and the 
next year it was $4120. How much greater was his income 
the second year than the first? 

75. ‘Two vessels start from the same point and travel in 
the same direction; the one 7821 miles, the other 5078 
miles. How far apart are they? 

76. A gentleman owns two vessels, one of which will 
carry 20763 feet of lumber, and the other 20976 feet. How 
much will the first vessel carry more than the second? 

77. A farmer dug two trenches, one of which will con- 
tain 11000 gallons of water, the other 8975 gallons. How 
many gallons does the latter contain less than the former? 

78. In the Chicago Water Works one of the engines 
will pump 38000000 gallons per day, another 18675450 gal- 
lons. How many gallons more does the larger engine pump 
than the smaller, per day? 

79. Gave $15234 for a farm, and $8796 less for the 
buildings upon it. How much did the buildings cost? 

SO. A gentleman having $55381, expended $16897 for 
cattle. How much had he left? 

S71. A man having an estate worth $78346, bequeathed 
to his two sons $49758. How much did he retain? 

52. A man bought a quantity of land for $12975, and 
sold it for $15250. How much did he gain? 


SUBTRACTION. 57 


$3. Illinois. contains 55405 square miles, and Indiana 
33809. How much larger is Illinois than Indiana? 

S54. Ohio contains 39964 square miles, and Oregon 
100000. How much less is Ohio than Oregon? 

Sd. The State of Ohio in 1870 raised 20539643 pounds 
of wool, and California 11391743 pounds. How much more 
was raised in Ohio than in California? 

56. The same year New York produced 1071: 


O26 
pounds of butter, and Pennsylvania 60834644 pounds. How 


much is the difference of their production? 

87. The population of Chicago in 1870 was 298977, and 
that of St. Louis was 310864. Which was the larger, and 
how much? 

S58. In 1870 the population of a certain city was 71440; 
it has increased so that the population now, 1876, is 83578. 
How much has been the increase? 

SI. In 1775 the war of the Revolution began, and in 
1861 began the Civil War of the U.S. How many years 
apart were they? 

90. The highest mountain in Asia is 29100 feet, and the 
highest in the United States is 17900 feet. What is the 
difference of their heights? 

91. New York in 1870 raised 652800 horses, and Illinois 
1008800. Which state produced the most horses. and how 
many? 

92. The taxes of a certain state were, in 1874, $9520100, 
and the following year $789890 less. How much were 
they in the following year? 

93. The receipts of a certain city for one year were 
$9223012, and the expenditures $8845764. How much were 
the receipts greater than the expenditures? 


WRITTEN PROBLEMS INVOLVING TWO OR MORE PROCESSES. 


94. A gentleman bought two farms; for one he paid 
$5476, and for the other $4562. He sold them both for 
$12580. How much did he gain? 


58 ELEMENTARY ARITHMETIC. 


95. Having $12847, I sold a house for $5489; how much 
more do I need so that I shall have $23425? 

96. A man worth $15300, gave $4360 to each of his two 
sons, and $4225 to his daughter, and the remainder to chari- 
table purposes. How much did he give in charity? 

97. Having $5895, I lost in trade $2750, then made 
$3960, afterward lost $4680, and then made $3690. How 
much did I have then? 

9S. From 11 million 300 thousand take the difference of 
ten thousand seventy and one hundred four. 

99. From 5 million 71 thousand five, take the difference 
of 2 hundred thousand 17, and 61 thousand 7, increased 
by their sum. 

100. Vice-President Wilson died in 1875 at the age of 
63. The Declaration of Independence occurred in 1776. 
How many years before the birth of Henry Wilson was 
Independence declared? 

101. A gentleman owning 589 acres of land, bought 
349 more; he afterward sold at one time 294 acres, and at 
another 57 acres. How many acres were left? 

102. Wf I travel Monday 2756 miles from home, Tues- 
day 1907 miles towards home, Wednesday 3874 miles from 
home, and Thursday return 2009 miles; how far from 
home am I Thursday night? 

103. A brickmaker had 2 kilns, each containing 43567 
bricks; he sold 8376 at three different times, and used 
12564. How many bricks had he left? 

104. London, the largest city in the world, had in 1870 
a population of 3082300, and New York, the largest city in 
America, a population of 942292. If, in a certain time, 
New York has an increase of 17058, and London 89856, how 
many more inhabitants will London have than New York? 

105. From a bin containing 80071 bushels of salt, a 
merchant sold to one man 1020 bushels, to another 129 
bushels, and to another the remainder. How many bushels 
did the last man buy? 


SUBTRACTION. 59 


106. An apple dealer in Chicago finds that out of 1849 
+956+987 barrels of apples sent to him from Michigan, 
280 barrels of them are damaged. How many orders of 800 
bbls. can be filled from the remainder? How many barrels 
will be left? 

107. Bought a house, lot, horse, and chaise, for $17800. 
If 1 paid $7890 for the lot, and $480 for the horse and chaise, 
how much was paid for the house? 
~ 108. If 11500500 and 12567658 be added to 7984307, the 
sum will be the number of inhabitants in France in 1830. 
The population of the British Empire at the same time was 
22297621. Which country had the larger population? and 
how many more? 

109. A speculator bought 3 houses, paying for the first 
$7010, for the second $13000, and for the third as much as 
for the first and second, less the number of dollars which 
the second cost more than the first. How much did he pay 
for the third house? . 

110. The source of the Missouri River is 6800 feet 
above the level of the sea; that of the Mississippi is 1680 
feet above the sea level. A certain spring is located 1009 
feet below the source of the Missouri. How far above or 
below the source of the Mississippi is it? 

111. 3 houses are found to be standing in a straight 
line; the second 5420 feet from the first, and the third 7337 
feet from the second. <A certain tree standing between the 
second and third is 538 feet from the third. How far is the 
tree from the first house? 

112. $6385 was paid for each of four houses, which 
were soon after sold so as to lose $987 on each of two of 
them. How much was received for the houses? 

113. A commission merchant received 7598 barrels of 
flour for each of six months, and sold 7438 for each of five 
months. How many barrels has he left? 

114. If I can save $568 a year from my income, how 
much less than $3600 can I save in 6 years? 


69 ELEMENTARY ARITHMETIC. 


115. Bought a house, farm, and 4 horses for $15310. Tf 
I paid $4375 for the house, and $560 for the horses, how 
much does my farm cost? 

116. A herds 19020 sheep, B 12780, and C 5460 less 
than A. How many sheep more or less does © herd than B? 

117. A,B and C, form a partnership with a capital of 
$25800; if their gains the first 5 years are respectively 
$1800, $2500 and $4875,.and their losses $980, $650 and 
$298, what are they all worth at the end of that time? 

118. An army of 100000 men engaged in battle, and 
was re-enforced by 29000 more; 237 deserted, 12888 were 
killed, 1500 were missing. How many were left in the 
army? 

119, The population of New York is about 4362834, of 
New Jersey 905794, of Pennsylvania 3519601, and of Dela- 
ware 125015. How much is the population of New York 
greater or less than the three other Middle States? 

120. In the year 1860 the number of teachers employed 
in the public schools of Illinois was 11099, and the number 
of pupils in attendance 433018; in Massachusetts the num- 
ber of teachers 5308, and of pupils 206974; in New York 
the number of teachers 15872, and of pupils 697273. How 
many more pupils than teachers in these States? 

121. Three brothers bought a factory, the first paying 
$5682.40, the second $8624.80, and the third as many dollars 
more than the second as the second paid more than the first. 
How much did the third brother pay? 

122. The brothers mentioned in the above example sold 
the factory for $30450, the first receiving $932.50 more than 
he paid, the second $1720.60 more than he paid, and the 
third the remainder. How much did the third brother 
receive? 

123. Aman purchased a house and farm for $7630.50; 
he expended $1230 in repairing the house, $987.50 in im- 
proving the farm, and $556.75 in planting an orchard, when 


he sold both for $12579. How much did he gain? 


SUBTRACTION. 61 


124. Bought a quantity of grain of one man for $1210.60, 
another quantity of another man for $929.85, a third quantity 
of a third man for as many dollars less than the second as the 
second is less than the first. How much was paid to the 
third man? 

125. Tf a man’s income is $200 a month, and _ his 
expenses are $35 a month for rent, $48.60 for grocery bills, 
and $45.75 for other expenses, how much can he save 1n 6 
months? 

126. The difference of $17846 and $27984, is what was 
paid for a certain property, it was soon after sold for $14025. 
How much was the gain? 

127. George Washington died in 1799 at the age of 67 
years. How long before his birth was the discovery of 
America in 1492? 

128. C has $15890; A has $1546 less than C. How 
many dollars has A? B has $6430 less than A; how many 
dollars has B? D has $62 more than A and B together; 
how many dollars has I)? 

129. A farmer filled 4 bins with grain. Into the third 
he put 2240 bushels; into the first, 420 bushels less than 
into the third; into the second, 840 bushels less than into 
the first; and into the fourth he put 650 less than into the 
first and second together. How many bushels did he put 
into each bin? 

130. A farmer being asked how many sheep he had, 
replied that if he had 420+4560 more, he should have 3628 
sheep. How many had he? 

131. After A, B, C, and D had traded for a year, it 
was found that C owned $21360, that A owned $930 less 
than C, that B owned $930 less than A, and that D owned 
$930 more than B. How many dollars did each own? 

132. A gentleman dying bequeathed his property, con- 
sisting of real estate to the value of $380500, and bank 
stocks amounting to $360500, as follows: to his brother 
$95640; to his sister as much as to his brother, less $15000; 


62 ELEMENTARY ARITHMETIC. 


to his son $59880 more than to his sister; to his wife as 
much as to the other three plus $18000; and the remainder 
to a charity. How much was given to charity? 

1353. A grain merchant in Chicago received two orders 
from New York, one for 15000, the other for 10000 bushels 
of corn, and a smaller order. In fulfillment of these orders 
he shipped 31780 bushels. How much was the smaller 
order? | 

134. From three times the sum of eighty-seven thou. 
sand six hundred four, forty-nine thousand three hundred 
fifty-eight, and twenty-nine thousand one hundred ninety, 
take four times the difference of the first two numbers, and 
tell how much the remainder exceeds the last of the given 
numbers? 

135. Bought 5 houses, each for $6788, and gave in pay- 
ment 6 lots, each for $3790, and the rest in money. How 
much money did I give? 

136. A revenue cutter cruising in the Pacific Ocean 
started from a certain point and sailed north 1525 miles, 
then south 825 miles, then north 728 miles, then south 960 
miles, then north 588 miles, then south 1438 miles, then 
north 980 miles, then south 840 miles. How far was the 
vessel from the starting point? In what direction? 

137. An aeronaut ascended from a certain point 3987 
feet during the first hour, descended 1876 feet the second 
hour, ascended 2019 feet the third hour, dropped down 
2569 feet the fourth hour, rose 3289 feet the fifth hour, 
sailed due east 12 miles the sixth hour, and then descended 
3429 feet the seventh hour. How far from the earth is the 
aeronaut at the end of the seventh hour? 


MULTIPLICATION. 63 


5 PG eLO Nerv: 


MULTIPLICATION. 


Art. 41, A farmer gave to each of his 4 sons 5 acres of 
ground. How much did he give them all? 


PROCESS OF ADDITION. 


5 acres+9) acres+5) acres+5 acres=20 acres. 


ANALysis.—Since the farmer gave to each son 5 acres, he gave to 
the 4 sons the sum of 5 acres++5 acres+-5 acres--5 acres=20 acres. 
The above is an illustration of examples in which it is required te 
obtain the sum of several numbers each equal to the other, and the 
method of obtaining this sum may be either by the above process 
called “process by addition,’ or by a much shorter process, as fol- 
lows: 
ANALysIs.—Since the farmer gave 
PRocEss BY MULTIPLICATION. to each son 5 acres, he gave to the 
4 times 5 acres=20 acres. four sons 4 times 5 acres=20 acres. 
This latter method is called the 
“process by multiplication.” Hence, 


42, Multiplication is a short method of obtaining 
the sum of several equal numbers. 


Thus, instead of saying 3+3-+3-++-3-+-3=15, we say 5 times 3=15, 
or 5 3’s=15. When the numbers to be added are both large and 
numerous, the process by addition would be very laborious. 

43, The terms employed in multiplication are multipli- 
cand, multiplier, and product. 

44, The Multiplicand is any one of the equal 
numbers to be added. 

Thus, in the problem above, 5 acres is the multiplicand. 


64. ELEMENTARY ARITHMETIC. 


45, The Multiplier is the number of equal num- 
bers to be added. 


Thus, in the same problem, 4 is the multiplier. 


46. The Product is the sum of the equal numbers to 
be added. 


Thus, in the same problem, 20 acres is the product. 


47, The Sign of Multiplication is xX, called 
times or multiplied by. When placed between two num- 
bers it indicates that one of the numbers is to be multiplied 
by the other. 

Thus, 3X5=15, is read 8 times 5 is 15, or 5 times 38 is 15; also, 
$4x5—$20, is read 5 times $4 is $20, or $4 multiplied by 5 is $20. 

48. The multiplicand and multiplier are called Factors 
of the product. 

Thus, 3 and 5 are factors of 15. 


PRINCIPLES. 


1. The multiplicand may be either an abstract or de- 
nominate number. 

2. The multiplier is always an abstract number. 

3. The product is always like the multiplicand. 

4. The multiplication or division of either factor by any 
number, multiples or divides the product by the same 
number. 


NOTE, — When both factors are abstract numbers, either may be used as the 
multiplicand or multiplier, but if one of these numbers is denominate, it must be 
considered the multiplicand. 


ORAL PROBLEMS. 


1. There are 4 quarts in 1 gallon. How many quarts 
are there in 5 gallons? ! 
GENERAL STATEMENT.—In 5 gallons there are 5 times as many 
quarts as there are in 1 gallon. 


ANALYsIS —Since there are 4 quarts in 1 gallon, in 5 gallons there 
are 5 times 4 quarts, which are 20 quarts. 


MULTIPLICATION. 65 


Therefore, in 5 gallons there are 20 quarts. 
NOTE. — It will be found a valuable exercise to require pupils to make a general 
statement of the solution of the problem. 
2. What will be the cost of 6 oranges, if one orange 
cost dc.? 
3. If one ton of coal cost $7, how much will 5 tons 
cost? 
4. ‘There are 8 quarts in one peck. How many quarts 
are there in 4 pecks? 
}. How many pecks are there in 9 bushels, if there are 
4 pecks in 1 bushel? . 
6. How much can a man earn in 6 days, if he can earn 
$8 in one day? 
7. A gentleman gave $12 to each of his 5 sons. How 
much did they receive? 
S. <A person traveled 12 miles a day for 7 days. What 
distance did he travel? 
9. A boy can earn $9 a month. How much can he 
earn in 9 months? 
10. There are 12 inches in one foot. How many inches 
in a pole 8 feet long? 
iI. There are. 3 feet in one yard. How many feet are 
there in 20 yards? 
12. How many inches are there in 2 yards? 
13. There are 2 pints in one quart. How many pints 
are there in 5 gallons? 
14. Bought 12 yards of broadcloth at $12 a yard. How 
much was the cost? 
15. Gave $6 a head for sheep, and $12 a head for hogs. 
How much will 5 sheep and 5 hogs cost? 
16. What is the cost of 12 pairs of boots at $8 a pair? 
17. Bought 9 yards of cloth at $5 a yard. How much 
change should I receive if I offer the trader $50. 
1S. A jady had a $50 note, a $20 note, and a $5 note. 
How much money had she left after buying 6 sheep at $12 


wh? 
each? “s 


66 ELEMENTARY ARITHMETIC. 


19, There are 16 ounces in one pound of tea. How 
many ounces in 5 pounds? 

20. Bought 6 oranges at 5c. each, and 5 lemons at 8c. 
each. How much did they all cost? 

21. Bought 12 yards of cloth, and 5 yards of velvet; 
the cloth at $5 a yard, and the velvet at $8 a yard. How 
much did I pay for both? 

22. How many days are there in 9 weeks? In 12 weeks? 
In 20 weeks? 

23. Built two fences; one 45 feet long, the other 12 
yards long. Which is the longer, and how much? 

24. There are 9 square feet in one square yard. How 
many square feet in a floor that contains 12 square 
yards? 

25. There are 4 quarts in one gallon. What is the cost 
of 3 gallons of oil at 12c. a quart? ; 

26. There are 3 feet in one yard. What is the cost of 
12 yards of carpet at $1 a foot? 

27. Since there are 4 pecks in one bushel, what is the 
cost of 5 bushels of peaches, at $2 a peck? 

28. Bought 5 tons of coal at $12 a ton, and 6 cords of 
wood at $5 a cord. How much was paid for both coal and 
wood? 

29, What is the cost of one gallon of molasses at 20c. a 
quart? 

30, Paid 75c. for a bushel of potatoes, and sold them at 
20c. a peck. How much did I gain or lose? 

G1. If a boy can earn $2 a day, how much can he earn 
in 6 weeks, omitting Sundays? 

32, A man paid $6 for a vest, twice as much for a pair 
of pants as for the vest, and twice as much for a coat as for 
the pair of pants. How much did he pay for the suit? 

33. Bought 3 boxes of raisins at $5 a box, and 5 boxes 
of lemons at $3 a box. How much did they both cost? 

34. Mary reads twice each day. How many times will 
she read in 4 weeks, of 5 days each? 


MULTIPLICATION. 67 


35, John counted 7 nines, and Mary 6 twelves. How 
many more did Mary count than John? 

36. Ina cornfield there are 8 rows of corn, having 9 hills 
in each row; in another there are 7 rows, having 7 hills in 
each row. If the crows tear up 2 hills in the first field, and 
3 rows in the second field, how many hills will remain? 
How many hills should be replanted? 

37. There are 8 furlongs in one mile. How many fur- 
longs are there in 9 miles and 6 furlongs? 

38. Since there are 12 inches in one foot, how many 
inches are there in 12 feet and 6 inches? 

39, A squirrel carried into his nest 5 acorns each day 
for 2 weeks. How many acorns did he gather? 

40. A farmer sold 5 tons of hay at $12 a ton, and took 
in exchange 7 barrels of flour at $8 a barrel, and the re- 
mainder in cash. How much cash did he receive? 

41. One farmer took to market 8 turkeys, weighing 12 
pounds each, and another took 12 geese, weighing 9 pounds 
each. How much more or less did the turkeys weigh than — 
the geese? 

42. Bought 12 yards of ribbon, at 12c. a yard, and gave 
in payment 120c. How much do J still owe? 

43. Two boys start from the same place and travel in the 
same direction, one at the rate of 5 miles an hour, the other 
at the rate of 9 miles an hour. How far apart will they be 
in 12 hours? 

44. Two men start from the same place and walk in 
opposite directions, one at the rate of 4 miles, the other at 
the rate of 5 miles an hour. How far apart will they be in 
12 hours? 

45. A farmer started to market with 8 bushels and 3 
pecks of wheat, and sold 5 bushels and 2 pecks to one man, 
and 3 pecks to another. How much wheat had he left? 

46. Bought 9 tons of coal at $12 a ton, and gave in pay- 
ment 12 barrels of flour at $8 a barrel, and the rest in cash. 
How much cash did I give? 


68 HLEMENTARY ARITHMETIC. 


47. Sold 12 yards of cloth at $7 a yard, receiving in 
payment 9 head of sheep at $8 a head, and the remainder 
in money. How much money did I receive? 

48. ‘Traveled on a journey afoot for 6 days at 12 miles a 
day, and returning rode 2 days at 25 miles per day, and 
walked the remaining distance home. How far did I walk on 
my return? How many miles did I walk on the journey? 

49, I have a box divided into 2 parts; in each part there 
are 3 parcels; in each parcel there are 4 bags; in each bag 
there are 5 marbles. How many marbles are there in the 
box? 

50. There are 4 farmers, each of whom has 5 fields of 
pasture; each field has 4 corners, and in each corner there 
are 9 sheep. How many sheep do the farmers own? 

O1. Since there are 4 pecks in one bushel, how many 
pecks are there in 9 bushels and 3 pecks? in 12 bushels and 
2 pecks? 

52. Since there are 12 inches in one foot, how many 
inches are there in 6 feet and 6 inches? In 9 feet 7 inches? 
In 12 feet 8 inches? 

53. Since there are 8 furlongs in one mile, how many 
furlongs are there in 7 miles and 5 furlongs? In 9 miles 
and 6 furlongs? In 12 miles and 7 furlongs? 

d4. A lady bought a gold chain weighing 2 pwt. 12 
grains. How many grains did it weigh, if there are 24 
grains in one pwt. 

55. Bought 5 ounces 15 pennyweights of gold, at $1 
per pennyweight. How much did it cost if there are 20 
pennyweights in one ounce? 

56. Sold 8 pounds 4 ounces of drugs at 9c. an ounce. 
How much was received for it if there are 12 ounces in one 
pound? 

57. Find cost of 12 gallons 2 quarts of oil at 10c. a 
quart; there being 4 quarts in one gallon. 

58. At 8 cents each, how much will 4 dozen and 2 lead 
pencils cost? 


MULTIPLICATION. 69 


59, A gentleman receives a salary of $72 a month, and 
his son a salary of $60 a month. How much more does the 
father receive than the son in 9 months? 

60. Bought 5 pounds 10 ounces of wool at 4c. an ounce. 


What is the cost if there are 16 ounces in one pound? 


ANALYSIS OF MULTIPLICATION. 


61. What is the product of 6 times 72? 


PRocESs BY ADDITION. Process BY MULTIPLICATION. 
02 02 
72 6 
72 — 
@2 432 
72 


ANALysI8s.—The analysis by addi- 
tion has been explained already, Art. 
41. The analysis by multiplication is 
432 as follows: 

1. For convenience write the mul- 


02 


tiplier beneath the multiplicand. 

2. Multiply each figure of the multiplicand by the multiplier, as 
follows: 6 times 2 units are 12 units, equal to 1 ten and 2 units. 
Write the 2 units in the order of units, and add the 1 ten to product 
of 6 times 7 tens, or 42 tens. 42 tens plus 1 ten are 43 tens, equal to 
4 hundreds and 3 tens. Write the 3 tens in the order of tens, and the 
4 hundreds in the order of hundreds. 

Therefore, the product of 6 times 72 is 482. 


62. What is the product of 46 times 3467? 
ANALYsIS.—1. Write 
the multiplier as before, 
3467 Multiplicand. beneath the multipli- 
46 Multiplier. cand, and multiply each 
—— figure of the multipli 
20802 Partial product, 34676. cand by each figure of 
13868 Partial product, 346740, the multiplier, begin. 
- ning at the order of 
159482 Entire product, 346746. units; thus, 6 times 7 
units are 42 units, equal 
to 4 tens and 2 units. Write the 2 units in the order of units, and 

add the 4 tenge to the product of 6 times 6 tens. 


PROCESS BY MULTIPLICATION. 


70 ELEMENTARY ARITHMETIC. 


2. 6times 6 tens are 36 tens. 36 tens plus 4 tens are 40 tens, equal 
to 4 hundreds and 0 tens. Write the 0 tens in the order of tens, and 
add the 4 hundreds to the product of 6 times 4 hundreds. 

3. 6 times 4 hundreds are 24 hundreds. 24 hundreds plus 4 hun- 
dreds are 28 hundreds, equal to 2 thousands and 8 hundreds. Write 
the 8 hundreds in the order of hundreds, and add the 2 thousands to 
the product of 6 times 2 thousands. 

4. 6times 3 thousands are 18 thousands. 18 thousands plus 2 
thousands are 20 thousands, equal to 2 ten-thousands and 0 thousands. 
Write the 0 thousands and the 2 ten-thousands in their respective 
orders. 

Therefore, 20802 the product of 34676 is a partial product of 
3467 X46. 

5. The second partial product is obtained by multiplying 3467 
by the 4 tens or 40 units; thus, 40 times 7 units are 280 units, equal 
to 2 hundreds and 8 tens. (The 0 units may be rejected.) Write the 
8 tens in the order of tens, and add the 2 hundreds to the proauct of 
40 times 6 tens. Proceeding as before, the second partial product, 
138680, is obtained. 

Adding the partial products, the entire product is found to be 
159482. 

Therefore, the product of 46 times 3467 is 159482. 

The same result is obtained by observing the following directions: 
Multiply each figure of the multiplicand by each figure of the multi- 
plier, writing the first figure of each partial product in the order 
occupied by that figure of the multiplier producing the partial pro- 
duct. 


PROOF OF MULTIPLICATION. 


I. Subtract the multiplicand from the product, and then 
from the remainder, and so continue until the number of 
subtractions equals the number of units in the multiplier. 

II. Division. Article 61. 


The continued product of several numbers is indicated by placing 
the sign X between each two of the numbers. 

Thus, 3X4X5X6=360, indicates the continued product of 3, 4, 5, 
6, which is obtained by multiplying one of the numbers by another, 


and the product by a third number, ete. 


NOoTE.—If, as sometimes happens, the multiplier is larger than the multiplicand, 
the latter, for convenience in practice, may be used as the multiplier. 


MULTIPLICATION. 71 


WRITTEN EXERCISES. 


68. Multiply the following numbers by each of the num-_ 


bers from 2 to 12 inclusive. 
Norer.—These exercises may serve to promote rapidity of execution. 


1.—58745 | 13.—900195 
2.— 63294 | 14.—354764 
3.—82563 | 15.—822078 
4.—42937 | 16.—323599 
).—04012 | 17.—765102 
6.—89645 | 18.—358455 
7.—d4785 | 19.—839768 
8.—49236 | 20.—467453 
9.— 36528 | 21.—370228 
10.— 73924 | 22.—995323 
11.—21045 | 23.—201567 
12.—54698 | 24.—554853 


20.—1967341 | 37.—20907683 
26.—4192093 | 38.—42765401 
27.—8765437 | 39.—22663973 
28.—9988776 | 40.—19977991 
29.—40359007 | 41.—83215946 
30.—2595139 | 42.—18671868 
31.—9611437 | 43.—73689202 
32.—3902914 | 44.—12345678 
33.—71856374 | 45.—91223344 
34.—6778899 | 46.—79911997 
39.—7129304 | 47.—64951238 
36.—9315925 | 48.—89012345 


Any integral number may be multiplied by 10, by 100, by 1000, 
etc., by annexing to the multiplicand as many ciphers as there are 


ciphers in the multiplier. 
Thus 34 x 100 = 3400. 


64, Find the products of the following factors by multi- 
plying the multiplicand by the significant figures of the 
multiplier, and annexing to the product thus obtained as 
many ciphers as there are on the right of the multiplier. 


1.—425 x 900 
2,—327 < 2400 
3.—d62 x 1800 
4,.—698 x 46000 
5.—849 x 57000 


6.—5468 x 12400 
7.—6275 x 30500 
8.— 840 x 29000 
9.—1504 x 20500 
10.—3700 x 56800 


65. tind the products of the following numbers: 


1.—463 x45 
2.—348 x 62 
—793 x 86 
4.—989 x 90 
5.—75 X42 x 56 
6.—84 x 37 X 69 
7.—7198 x 256 
8.—93186 x 445 
9,— 99999 « 999 
10.—76854 x 800 


11.—90763 x 700 
12.—7422153 x 468 
13.—6929867 x 5000 
14.—9507340 x 7071 
15.— 9264397 x 4762 
16.—1534693 x 9584 
17.—9999999 x 9999 
18.—3854 x 3854 x 3854 
19.—1428576 x 70000 
20.—7050860 x 70508 


72 -ELEMENTARY ARITHMETIC. 


66. Multiply 657142 by each of the numbers from 56 to 
81 inclusive. 

67. Multiply 768354 by each of the numbers from 64 to 
89 inclusive. 

6S. Multiply 876543 by each of the numbers from 78 to 
103 inclusive. 

69, Multiply 98998 by each of the numbers from 359 to 
384 inclusive. 


WRITTEN PROBLEMS. 


. 


70. There are 63 gallons in one hogshead. How many 
gallons are there in 25 hogsheads? 


PROCESS. SoLuTION.—1. Sfatement——In 25 hogsheads there 
63 gallons. are 25 times as many gallons as there are in 1 
25 hogshead. 

2. Analysis—Since there are 63 gallons in one 

815 hogshead, in 25 hogsheads there are 25 times 63 gal- 

126 lons, which are 1575 gallons. 


3. Conclusion.—Therefore, in 25 hogsheads there 
are 1575 gallons. 


1575 gallons. 


71. How many gallons are there in 36 hogsheads and 36 
gallons? 

72. How many gallons are there in 72 hogsheads and 54 
gallons? 

73. What is the cost of 48 hogsheads and 32 gallons of 
wine, at $3 a gallon? 

74. There are 365 days in one year. How many days 
did Vice President Wilson live, who died at the age of 63 
years? 

75. A certain man arose at-5 o’clock each morning for 
35 years; another arose at 7 during the same period. How 
many hours did the former gain upon the latter? 

76. The average earnings of a certain gentleman during 
a period of 48 years was $6 per day. What were his re- 
ceipts during that time? 


MULTIPLICATION. 73 


77. There are 320 rods in one mile. What is the cost 
of 75 miles of road bed, at $75 a rod? 

78. What is the cost of 176 miles of railroad, at $5485 
a mile? 

79. A gentleman purchased a farm of 468 acres, at $39 
an acre, and soon after sold it at $47 an acre. How much 
did he gain? 

SO. Bought 175 head of horses at $127 each; 96 head 
of cattle at $56 each, and 328 hogs at $19 each; they were 
afterwards sold at a gain of $956. How much was received 
for them? 

S71. There are 2000 pounds in one ton of hay. How 
many pounds of hay can be cut from 12 acres of meadow 
that yields 3 tons an acre? 

$2. How much is the hay mentioned in the preceding 
example worth at $18 a ton? 

$3. Bought 356 acres of land, 237 acres of which is 
under cultivation, and worth $37 an acre, and the remainder 
is woodland, worth $29 an acre. I gave in part payment a 
house and lot worth $12150. How much is still owing? 

$4. Bought 12 house lots at $1680 each, and erected as 
many houses at a cost of $3875 each. What was the whole 
cost? 

S55. Purchased a farm of 360 acres, at $35 an acre, and 
sold 196 acres, at $36 an acre, and the remainder at $33 an 
acre. Did I gain or lose, and how much? 

S56. Bought a drove of 225 cattle at $25 a head; their 
cost of transportation was $1125. I afterwards sold them 
at $37 a head. How much did I gain? 

87. Shipped 18 car loads of hogs, containing 28 hogs 
each, which I sold at $16 apiece. How much did I receive 
for them? 

88. Bought a cargo of coal consisting of 378 tons, at 
$6.50 a ton, and sold 217 tons at $7.40, and the remainder 
at $8 a ton. How much do I make? 

89. A gentleman’s income is $9875.50 a year, and his 


74 ELEMENTARY ARITHMETIC. 


expenses a month are as follows: rent $125, household 
economy $135, and all other expenses $230. How much 
can he save each year? 

90. If agentleman’s salary is $4000 a year, and he pays 
$36 a month for board, $198 a year for clothes, and $475 
for all other expenses, how much can he save in 4 years? 

91. There are 5280 feet in one mile. What is the cost 
of a railroad 168 miles and 3600 feet in length, at $3 a foot? 

92. Purchased 9 lots at $1875 each, and erected upon 
them as many houses, at a cost of $4288 each. They were 
sold for $70000. What was my gain? 

93. My bank stock consists of 4 $1000 bonds. All my 
other property, in cash, is worth 6 times as much as my 
bank stock. If I buy 358 acres of land at $36 an acre, 
how much will I have left? 

94.. A gentleman started upon a journey of 2000 miles, 
at the rate of 36 miles a day. After traveling 7 weeks and 
4 days, omitting Sundays, what is the distance yet to be 
traveled? 

95. In a certain army the number that came out of the 
battle was 12 times the number killed, which was 6436. 
The number wounded and missing was 3 times the number 
killed, less 9439. How many men went into the engage- 
ment? 

96. A gentleman in making a journey around the world 
found that the distance traveled by vessel was 19 times as 
far as that traveled by rail, which was 1360 miles, and the 
distance by other conveyances was equal to that traveled 
by rail, less 1065 miles. What was the whole distance 
traveled? 

97. What is the cost of 312 bushels of nuts, at 26c. a 
quart? 

98. Find the cost of 78 hogsheads of wine, at 56c. a 
pint. 

99. <A clock ticks 3600 times in one hour. How often 
does it tick in 8 days and 3 hours? 


MULTIPLICATION. T5 


JOO. In one cubic foot there are 1728 cubic inches. 
How many cubic inches in 96 cubic feet and 1245 cubic 
inches? 

LO1. How far will a locomotive travel in 2 weeks, omit- 
ting Sundays, at the rate of 28 miles an hour for 8 hours 
each day? 

102. What is the cost of 1675 chests of tea, each chest 
containing 67 pounds each, at $1.85 a pound? 

103. Purchased 475 bushels of wheat at $1.58 a bushel, 
96 bushels of corn at 56c. a bushel, and 157 bushels of oats 
at 65c. a bushel. What was the cost of all? 

104. Bought a bill of goods as follows: 27 yards of cloth 
at $8.25 a yard, 56 yards of velvet at $9.80 a yard, 36 yards 
of silk at $2.75 a yard, and 125 yards of cotton cloth at 9c. 
a yard. How much was the cost of the goods? 

105. How many minutes longer is the month of July 
than the month of June? How many minutes are there in 
June, July and August? 

106. A house is worth $2450, the farm on which it 
stands is worth 12 times as much as the house, less $600, 
and the stock is worth twice as much as the house. What 
is the value of the house, farm and stock? 

107. If a person receives an annual salary of $1875, 
and expends each year $312 for board, $105 for clothing, 
and $275 for other purposes, how much will he save in 18 
years? 

108. Purchased 58 tons of coal at $7.50 a ton, 96 cords 
of maple wood at $5.75 a cord, 128 cords of pine wood at 
$4.80 a cord. If I pay $1296 cash, how much remains to 
be paid? 

109, Paid $136 for a wagon, 3 times as much for a yoke 
of oxen as for the wagon, and twice as much as for the oxen, 
less $265, for a span of horses. How much did they all 
cost? 

110. A drover bought 356 beeves at $35 a head, 1235 
sheep at $3.50 a head, and paid for their transportation to 


76 ELEMENTARY ARITHMETIC. 


market $975; he afterwards sold both the beeves and sheep 
for $21360. How much did he gain? 

111, A merchant bought 63 pounds of sugar at lle. a 
pound; 6 chests of tea, containing 78 pounds each, at $1.47 
a pound; 2 hhd. of molasses, 63 gallons each, at 30dc. a 
‘gallon; and 5 barrels of oil, 31 gallons each, at 28c. a gal- 
lon. What was the cost of the whole bill? 

112. Bought 45 cases of boots, each containing 15 pairs, 
at $5.80 a pair; 36 cases of shoes, 24 pairs each, at $3.90 a 
pair; 24 cases of gaiters, 16 pairs each, at $3.25 a pair; and 
56 pairs of slippers at $2.25 a pair. How much was the 
whole cost? 

118, Bought a house, farm and the stock thereon; for the 
house $2197 was paid; the farm cost 5 times as much, less 
$1979, and the stock cost as much as the house and farm, 
less $4976. How much did they all cost? 

114. A is worth $5466; B 7 times as much, less $1387; 
and C 3 times as much as A and B, less $2348. How much 
are they all worth? 

115. If aman breathes 16 times a minute, how often 
will he breathe in one day, if there are 60 minutes in one 
hour, and 24 hours in one day? 

116. What is the cost of 12 pounds 8 ounces of tea at 
llc. an ounce, if there are 16 ounces in a pound? 

117. What is the cost of a beam whose length is 12 
yards 2 feet, at 75c. a foot? 

1185, There are 20 quires in one ream, and 24 sheets in 
one quire. What is the cost of 12 reams, 13 quires, and 17 - 
sheets of note paper, at 2c. a sheet? 

119, There are 100 pounds in one cwt. What is the 
cost of 18 cwt. 75 pounds of beef, at 6c. a pound? 

120. Mr. B bought at one time 758 acres of land at 
$87.56 an acre, and at another time 96 acres at $67.48 an 
acre; he afterwards sold it all at $75.85 an acre. Did he 
gain or lose, and how much? 

121. A capitalist who owned $2985 worth of bank stock, 


MULTIPLICATION. te 


and a farm worth 8 times as much, traded both for a factory 
worth $29750. How much remained to be paid? 

122. A gentleman built 12 houses at a cost of $4725 
each, and sold 8 of them at $5192 each, and the remainder 
at $4180 each. Did he gain or lose, and how much? 

123, A steamboat travels 12 miles an hour. How far, at _ 
that rate, will the same boat travel in one week? 

124, A merchant bought 45 pieces of cloth, each con- 
taining 28 yards, at $8 a yard. What was the cost? 

125, A planter sold 125 bales of cotton, each weighing 
345 pounds, at 27c. a pound. How much did he receive 
for the cotton? 

126. A miller sold 650 barrels of flour, each weighing 
196 pounds, at 5c. a pound; he took in part payment a yoke 
of oxen worth $250, and the rest in cash. How much cash 
did he receive? 

127. A farmer owns 3 farms; each farm contains 165 
acres; each acre produced 27 bushels of wheat, worth $2 a 
bushel. How much was the whole crop worth? 

128. A bought of B 563 acres of land, at $48 an acre, 
and gave in payment.a house worth $3756; a factory worth 
6 times as much, less $1267, and the rest in money. How 
much money did A pay? 

129. A horse is worth $96; the field in which he is 
pastured is worth 12 times as much; the whole farm is 
worth 9 times as much as the field; and the house $3216 
less than the farm. How much is the house worth? 

130. A is worth $2798; B 7 times as much, less $3289; 
(© 12 times as much as B, plus $976; and D 8 times as much 
as A and B. How much are they all worth? 

IG1. ¥., L. & Co. bought 85 pieces of silk, containing 48 
yards each, for $10200, and sold it at $4 a yard. How many 
dollars was gained? 

182. My house cost $1825; my farm cost $1200 more than 
5 times the cost of my house. What is the cost of the 
house and farm? 


78 HLEMENTARY ARITHMETIC. 


3 Cole @uNiwe Vids 


DLS LON. 


Art. 49, 1. A gentleman divided 24 acres of land 
among his sons, giving to each son 6 acres. How many 
sons had he? 


In this example it is proposed to divide 24 acres into a certain 
number of equal parts, each part of the size of 6 acres. 

Since he divided 24 acres so as to give each son 6 acres, it is evi- 
dent that there are as many sons as their are 6 acres in 24 acres. In 
24 acres there are 4 times 6 acres. 

Therefore, he had 4 sons. 


50. 2%. A gentleman divided 24 acres of land equally 
among his 4 sons. How many acres did each son receive? 

In this example it is proposed to divide 24 acres into 4 equal parts, 
so as to find the size of each part. 

Since he divided 24 acres of land into 4 equal parts, each part is 
of that size of which it requires 4 of them to equal 24 acres; each of 
those parts is therefore one fourth of 24 acres, which is 6 acres; there- 
fore, each son received 6 acres. 


51. The first of these examples illustrates a form of 
the division of a quantity into a certain number of parts of 
a known size, and it will readily be perceived that this form 
of division is, in reality, only a short method of finding the 
remainder obtained by the continued successive subtractions 
of a series of equal numbers from some given number; as 
multiplication is a convenient method of finding the sum of 
several equal numbers. 


For instance, in first example deduct one 6 acres, given to one son, 
and the remainder is 18 acres; in like manner deduct another 6 acres, 


DIVISION. 79 


given to another son, from 18 acres, and the second remainder is 12 
acres: again deduct a third 6 acres, given to a third son, from 12 acres, 
and the third remainder is 6 acres, which, given to the fourth son, 
leaves no remainder, and the number of times the known part, 6 
acres, is deducted from the whole quantity, is the required number of 
parts. 

It sometimes happens that the last remainder is greater than 0, but 
it is always less than the constant subtrahend. 

Thus, 22—4=18; 18—4=—14; 14—4=—10; 10—4=6; 6—4=2; show- 
ing there are five 4’s in 22, with a remainder, 2. 

52, The second of these examples illustrates a form of 
the division of a quantity into a given number of equal 
parts, so as to ascertain the size of the parts. 

Hence, Division is two-fold, as follows: 

53. 1. Division isa process of finding the number 
of equal parts or numbers, into which a given number is 
separated, the size or part being known. 

Thus, if 12c. be divided among a number of boys, giving each boy 
3c., how many boys are there? 

Here the whole quantity and the size of the parts are given to find 
the number of equal parts. 

2. Division is a process of separating a given number 
into a required number of equal parts. 

Thus, if 12c. be divided equally among 4 boys, how many cents 
will each boy receive ? 

_ Here the whole quantity and the number of equal parts are given 
to find the size of each part. 


Nore.—In all written operations the actual process in both forms of division is 
identical, though the forms of analysis of all examples in both oral and written 
problems are dependent upon the peculiar nature of the example. 


54, The terms used in division are dividend, divisor, 
and quotient. 

55. The Dividend is the number which contains a 
certain number of equal parts of a known size; or it is the 
number to be separated into a given number of equal parts. 


Thus, in the first and second examples 24 acres is the dividend. 


56. The Divisor is the number of equal parts or 
numbers into which the dividend is to be separated; or it is 


80 ELEMENTARY ARITHMETIC. 


one of the equal parts or numbers into which the dividend 
is to be separated. 

Thus, in the first example 6 acres is the divisor, and in the second, 
4 is the divisor. 

57, The Quotient, in the first form of division, is the 
- number of times the divisor is contained in the dividend; 
in the second form it is the value of one of the equal parts 
into which the dividend is to be separated. 

Thus, in the first example, 4 is the quotient, and in the second, 6 
acres is the quotient. 


58, The Remainder, in division, is that part of the 
dividend which remains after separating it into a certain 
number of equal parts. 

59. The Sign of Division is +. When placed 
between numbers it signifies that the first number is to be 
divided by the second. 

Thus, 36+6=6 is read 36 divided by 6 equals 6. 


Division is also indicated by writing the dividend above, 
and the divisor below, a horizontal line; or by writing the 
divisor on the left, and the dividend on the right, of a 
vertical or curved line. 

Thus, 32—6, 6|86=6, 6)36=6. 

It will be noticed that in the second form of division the divisor 
denotes one of the equal parts into which the dividend is separated. 
The name of the parts being determined by the number of units in- 
the divisor. 

Thus, if the dividend be divided into two equal parts, each of the 
parts is called one half of the dividend; if into three equal parts, each 
part is called one third of the dividend; if into e¢ght equal parts, each 
part is called one edghth of the dividend, ete. 

The one half of a number is found by dividing the number by two ; 
the one third of it is found by dividing it by three, and so on. In 
general, a given part of a number is found by dividing it by the 
number indicating the name of the parts. 

60. In division, the divisor and quotient are factors of 
the dividend; as in multiplication the multiplicand and 
multiplier are factors of the product. It will be perceived, 


DIVISION. 81 


therefore, that division is the converse of multiplication; in 
the latter the two factors are given to find the product; 
while in the former the dividend, corresponding to the pro- 
duct, and the divisor, corresponding to one of the factors, 
are given to find the quotient, corresponding to the other 
factor. 

61, Division and multiplication are proofs of each 
other. Zhe product divided by a factor equals the other 
Jactor; or the product of the divisor and quotient equals the 
dividend. 

PRINCIPLES. 


1. The dividend may be abstract or denominate. 

2. If the dividend is denominate, either the divisor or 
quotient is denominate, but not both. 

3. If the dividend is abstract, both divisor and quotient 
are abstract. : 

4, The remainder is always like the dividend. 


ORAL PROBLEMS. 


3. There are 4 quarts in one gallon. How many gallons 
are there in 32 quarts? 

STATEMENT.—There are as many gallons in We given number of 
quarts as the number of quarts in one gallon is contained times in 
the given number of quarts. 

ANALYsIs.—Since there are 4 quarts in one gallon, there are as 
many gallons in 32 quarts as 4 quarts are contained times in 32 
quarts, which are 8 times. 

ConcLuston.—Therefore, in 32 quarts there are 8 gallons. 


4. If $32 be equally divided among 4 persons, how 
much will each one receive? 

STATEMENT.—Each person will receive such a part of the whole 
number of dollars as each person is a part of the given number of 
persons. 

ANALYsIS.—Since $32 is divided equally among 4 persons, each 


person will receive ¢ of $32, which is $8. 
ConcLusion.—Therefore, each person will receive $8. 


= 


82 ELEMENTARY ARITHMETIC. 


5. If $63 was paid for 9 yards of cloth, what is the 
cost of one yard? 

6. A gentleman paid $72 for 12 head of sheep. What 
is the cost of one sheep? 

7. There are 8 furlongs in one mile. How many miles 
are there in 72 furlongs? 

S. Paid 96c. for 12 yards of ribbon. What is the cost 
of one yard? 

9. If a man can dig 84 feet of trench in 7 days, how 
many feet can he dig in one day? 

1O. If 5 yards of silk cost $15, how much will 8 yards cost? 

11. If 6 men can do a piece of work in 12 days, how 
long will it require 8 men to do the same? 

12. How much will 7 yards of cloth cost if 8 yards cost 
$56? 

13, What will be the cost of 20 oranges, if 9 oranges 
cost 45c.? ) 

14. How many tons of coal at $9 a ton will pay for 12 
barrels of flour at $6 a barrel? 

15. A gentleman bought 15 yards of velvet at $4 a yard, 
and gave in payment $12, and the remainder in flour, at $6 
a barrel. How many barrels did he give? 

16. There are 12 ounces in one pound. How many 
pounds are there in 108 ounces? ) 

17. If 8 quarts of molasses cost 72c., how much will 4 
of 15 quarts cost? 

18. What must be paid to ride +45 of 96 miles, if it cost 
d6c. to ride 7 miles? 

19. If 9 men can mow 36 acres in one day, how much 
can 12 men mow? 

20. If 6 men can build a boat in 14 days how long will 
it require 7 men to do it? 

21. If 9 pounds of sugar cost 108¢., how much will $ of 
a pound cost? 

22. Wow much will 9 barrels of cider cost, if 5 barrels 
cost $454 


~ bushels cost $40? 


DIVISION. 85 

23. Gave $63 for 7 tons of coal. How much should be 
paid for 12 tons? 

24. Bought 80 ounces of tea at $2 a pound. What is 
the cost? 

25. Gave 4 of $48 for 4 yards of silk. How much will 
4 of 24 yards cost, at the same rate? 

26. If 12 boxes of oranges cost $72, how much will 8 
boxes cost? 

27. Paid 80c. for 4 quires of paper. How much should 
be paid for 9 quires? 

28. If 6 men can do a piece of work in 15 days, in what 
time can 9 men do it? 

29. How many men can do as much in 6 days as 5 men 
can in 12 days? 

30. How many men can build a wall in 10 days, if 5 men 
can build it in 40 days? 

31. What is the cost of $ of 16 yards of cloth, if 6 
yards cost $48? 

32. If 8 acres of land cost $320, how much will 12 acres 
cost? 

33. What is the cost of 4 of 15 pounds of coffee if 4 
pounds cost 80c¢? 

SUGGESTION.— #=twice as much as}. 4of 15 is 5, $ is two times 
5, which are 10. 

34. How much will ¢ of 12 yards of cashmere cost if 12 


* yards cost $36? 


35. Tf 8 men can dig a trench in 20 days, how many 
men will be required to do it in 16 days? 

36. What will be the cost of 9 yards of cloth if 7 yards 
cost $5.60? 

37. What is the cost of 2 of 16 bushels of wheat if 20 
38. If 4 of 15 yards of ribbon cost 84c., how much will 


4 of 18 yards cost? 
39. Bought 9 barrels of flour for $72, and gave 6 barrels 


- for coal at $6 a ton. How many tons did I receive? 


84 ELEMENTARY ARITHMETIC. 


40. % of $40 is what I gave for 5 yards of cloth. What 
is the price of one yard? 

41. § of $40 is $5 less than what was paid for 8 sheep. 
What did one sheep cost? 

42, Paid $60 for 12 barrels of flour. At what price per 
barrel should it be sold so that $12 may be gained? 

43. I sold 12 dozen of pens for 84c, which was at a loss 
of 12c. How much did they cost per dozen? 

44. What is the cost of } of 18 sheep if ? of 12 sheep 
cost $63? 

4). A wagon cost $60, which is 5 times the cost of the 
harness. What is the cost of both the harness and wagony 

46. A watch cost $120, which is 4 times the cost of the 
chain. What is the cost of both watch and chain? 

47. Paid $96 for a sleigh, which was 8 times the cost of 
the bells, less $5. What was the cost of the bells? Of 
both? 

48. + of $60 is $3 more than what was paid for 9 barrels 
of salt? What was the salt a barrel? 

49, Tf 8 pounds of coffee cost $1.60, how much will ? 
of a pound cost? 

50. A tree is 72 feet high, which is 9 times the distance 
around it at the base. How many feet around the base? 

51. Bought 12 yards of velvet at $9 a yard, and sold it 
at a gain of $24. What was the selling price a yard? 

52, Paid $75 for 25 sheep, and soon after sold them ata - 
gain of $25. How much was received for each? 

53. Bought at one time 8 barrels of flour at $7 a barrel, 
and 4 barrels at $8 a barrel, and sold both lots so as to gain 
$20. What was the selling price a barrel? 


ORAL COMBINATIONS FOR DICTATION. 


b4. 12X%74+6+94124+5+3x44+8+4+5+7xK5—15=? 
HDs 9968692 7 19 523 a 9 eed eae 
56. 56=-7+-12—9 x 7—74+-11+9«K64+2+8x7—19 =? 
57. 682-9 xX 61964-19499 ee ee ee 


DIVISION. 85 


dS. 35413+6x«12+4+10x844+7x12—44+10=? 
59, 9XT—7T+74+-7X*3415+5 X9412+12x5+2=? 
60. 154+9+4x8—12+4x84+3+3x4—5046+7=? 
61. 18+12+5x9412+6x5—15+5x7—16—10=? 


62, 27+ 


38—9 K124-18+2 x 9—9+9 x54+16+7x8=? 


68. 367-12—8=—8-+10x4—12+8«12+4-3-~3x4=? 
G4. 385-15 x44+4+7x4+6x«8—4+5x3—16=? 

65. meee Ca Oo oie — 1 eres 

66. 382—12+20«x2+8 x6412+12x7—1246+3= 

67. 9494-9~3«7—7~+7x«5+20+-5x«4—18=? 

68. 8+84+8+8~—4x9+6x11—32~—10x8+4+7=? 
Bae ole 9 2 te LL el A ee = 165 eA oe 
70. %1124+6+8+-7-+-9+114+12—8—7—6—9+6=? 
og rE ay ge SS a a ee a pees eg 
72 12-+-18—5+16+8-+-12~6 k7-+-11+5 *8—15=? 
"Ei SIGE Sb Ses oe Ge Res pay = Ss ts eo baie ae 
74. 144+124+124124+12+6x748+8x549+7=? 
Pr ee 6 x Oe & 6 1S 6S? 
76. 108+94+7474+74+7+5 x9+12x10+40—50=? 
7. 50+50+100+200—300+10x74+-7+7+7x6=? 
78. 20+20+40+16+12 x 7—16~—8+95+21+11=? 


ANALYSIS OF DIVISION. 


79. Divide 2563 by 6. 


PROCESS. 
6)2563(42 fi 4 
24 


16 
12 


ANALYsIs.—1. Write the divisor at the left of the 
dividend, with a curved line between them, and for 
convenience begin to divide at the highest order of 
the dividend, thus: 

2. 6 is contained in 2 thousands 0 thousands times, 
with a remainder of 2 thousands, which change to 
hundreds. 2 thousands equal 20 hundreds. 20 hun- 
dreds plus 5 hundreds equals 25 hundreds. 

3. 6 is contained in 25 hundreds 4 hundred times, 
with a remainder of 1 hundred, which change to tens, 
1 hundred equals 10 tens. 10 tens plus 6 tens equals 
16 tens. 

4. 6 is contained in 16 tens 2 tens times, with a 


86 HLEMENTARY ARITHMETIC. 


remainder of 4 tens, which change to units. 4 tens equals 40 units 
40 units plus 3 units equals 48 units. 

5. 6 is contained in 43 units 7 units times , with a remainder of 1 
unit. Since 1 unit is less than 6, the “division can only be indicated 
thus, +. 

Hence, 2563+6=4274. 


The above analysis illustrates that form of division in 
which it is required to find how often one number is con- 
tained in another of the same kind. 

The following will illustrate the division of a number 
into a certain number of equal parts, and for this purpose 
it is assumed to find the 4 of 2563, the written process 
being identical in both cases; thus, 

Process. PRooF. ele —1. Write the numbers as before. 
2. ¢ é ‘ 
6 ) 2563 (427 nai S Rceeaan es ae ie 
24 6 2 thousands equals 20 hundreds. 20 hundreds 
ae ae: pans equals 25 hundreds. 
16 2562 3. 4 of 25 hundreds is 4 hundreds, with a re- 
12 1 mainder 1 hundred, which change to tens. 1 


hundred equals 10 tens. 10 tens plus 6 tens 
43 2563 — equals 16 tens. 


42 4. 4 of 16 tens is 2 tens, with a remainder of 
te 4 tens, aeh change to units. 4 tens is equal to 
1 40 units. 40 units plus 3 units equals 43 units. 


5. 4% of 43 units is 7 units, with a remainder 
of 1 unit. 


Hence, § of 2563 is 427, with a remainder 1. 


The preceding operations are by a process called “ Long 
Division.” The following will serve to illustrate the 
same operation by a process called “Short Division,” 
which differs from the preceding in that the product of the 
divisor by each succeeding figure of the quotient, and the 
successive subtractions of these products are not written. 

Thus, 6)2563 


4074 


NoOTE.—Operations in short division are usually confined to those examples in 
which the divisor does not exceed 12. 


DIVISION. 


WRITTEN EXERCISES. 


87 


SO. Divide the following numbers by each of the num- 


bers from 2 to 12 inclusive. 


1.—54698 
2.—21045 
3.—42937 
4.—82563 
5.— 63294 


6.—d8745 
7.—04698 
8.—21045 
9.—73924 
10.—36528 


11.—49236 
12.—54785 
13.—78545 
14.—36294 
15.— 92437 
16.—69458 
17.— 65283 
18.—45210 
19.—64985 
20.—82863 


21.—590019 
22.—467453 
23.—370228 
24.—995323 
25.—201567 
26.—954835 
27.—863978 
28.—394764 
29.—822073 
30.—323599 
31.—765102 
32.—358455 
33.—745346 
34.—915009 
30.—674534 
36.—455853 
37.— 702028 
38.—746053 
39.—65 1027 
40.—807320 


41.—1457691 
42.—3902914 
43.—7345678 
44,.—6778898 
45.—7009304 
46.—9315952 
47,.—7341169 
48.—4192093 
49.—4736587 
50.—9988776 
d1.—4039217 
D2.—)295139 
93.—7341196 
54.—9209314 
55.—5437678 
56.—8776988 
57.—9007304 
58.—5139592 
59.—4371167 


60.—5637478 | 


61.—38670902 
62.—10456724 
63.—37 936226 
64.—19977991 
65.—64951238 
66.—86817681 
67.— 20290753 
68.—87654312 
69.— 44332219 
70.— 79911997 
71.—83215946 
72.—54321098 
73.— 65431287 
74.—29075248 
75.—49512386 
76.—93626372 
77.— 67093026 
78.—81768186 
79.—90752482 
80.—36263726 


FIND THE QUOTIENTS OF THE FOLLOWING NUMBERS. 


S51. Divide 47836 by each of the numbers from 13 to 19, 


inclusive. 


S2. Divide 75048 by each of the numbers from 20 to 29, 


inclusive. 


85 Divide 


inclusive. 


8&4. Divide 


49, inclusive. 
85. Divide 
59, inclusive. 
86. Divide 


69, inclusive, 


93840 by each of the numbers from 30 to 39, 
325000 by each of the numbers from 40 to 
421648 by each of the numbers from 50 to 


356405 by each of the numbers from 60 to 


88 ELEMENTARY ARITHMETIC. 


87. Divide 483706 by each of the numbers from 70, 
to 79, inclusive. 

&8. Divide 563848 by each of the numbers from 80 
to 89, inclusive. 

&9. Divide 7056845 by each of the numbers from.90 
to 99, inclusive. 

90. Divide 8325436 by each of the numbers from 100 
to 109, inclusive. 

91, Find $ of 346228; 453764; 718296; 2937573. 

92, Find t of 423675; 5 576840; 68 id 8536410; 

701275; 874600; 965435; 1287595. 
93. Find 4 of 732848; 687256; 84362 g. 9427638; 


304982; 464544; 578976; 874967 
94. Find 4 of 385363; 785449; 679384; 8765477; 
499375; 589754; 691866; 7082943 
95. Find 4 of 784684; 783627; 874654; 9763481; 
587816; 679436; 738848; 9353976. 
96. Divide 543284 by 128; by 156; by 189. 
97. Divide 6378424 by 236; by 352; by 475. 
98. Divide 4863745 by 375; by 468; by 584. 
99, Divide 7548638 by 488; by 557; by 639 


100. Divide 8376547 by 654; by 729; by 856. 

101. Divide 6745438 by Baa of the numbers from 371 
to 390, inclusive. 

102. Divide 7438456 by each of the numbers from 452 
to 470, inclusive. — 

103. Divide 8436782 by each of the numbers from 654 
to 667, inclusive. ca, 

104. Divide 5337846 by each of the numbers from 829 
to 842, inclusive. 

105. Divide 6007408 by each of the numbers from 947 
to 965, inclusive. 


WRITTEN PROBLEMS. 


106. There are 24 grains in one pennyweight of gold. 
How many pennyweights are there in 864 grains? 


DIVISION. 89 


PROCESS. SoLtutrron.—1. Statement. There are as many penny- 
24) 864 (36 weights in the whole number of grains as the number 
) ( of grains in one pennyweight is contained times in the 


72 whole number of grains. 

aes 2. Analysts. Since there are 24 graius in one penny- 

144 weight, in 864 grains there are as many pennyweights 

144 as 24 grains is contained times in 864 grains, which are 
36 times. 


3. Conclusion. Therefore, in 864 grains there ure 36 pennyweights. 
107. Paid $1225 for 25 head of cattle. What was the 
cost of one head? 
PROCEss. SoLution.--1. Statement. The cost of one head 
25)$122 3(849 is such a part of the whole cost as one head is a part 
of the whole number of cattle. 
100 2. Analysts... Since 25 head of cattle cost $1225, 


295 one head costs lz of $1225, which are $49. 
3. Conclusion. Therefore, one head of cattle costs 


225 $49. 

LOS. ‘here are 20 pennyweights in one ounce of silver. 
How many ounces are there in 1260 pennyweights? 

JO9, There are 32 quarts in one bushel. How many 
bushels are there in 2496 quarts? 

110. Gave $3321 for 27 acres of land. What is the 
price per acre? 

J11, Traveled 1088 miles in 28 days. What was the 
distance traveled per day? 

112, A gentleman paid $4500 for 36 horses. What is 
the price of each horse? 

115, There are 63 gallons in one hogshead. How many 
hogsheads are there in 12474 gallons? 

114. How many years are there in 11700 weeks, if there 
are 52 weeks in one year? 

115, How many cubic yards are there in 6912 cubic feet, 
if there are 27 cubic feet in one cubic yard? 

116. If there are 320 rods in one mile, how many miles 
are there in both sides of a roadway 6080 rods in length? 

117. If each rail of a railroad track is 30 feet in length, 
how many rails will be required to lay a track 5280 feet? 


90 ELEMENTARY ARITHMETIC. 


118. 3 times $16000 is what was paid for 12 houses. 
What is the cost of each house? What is the cost of 7 
houses? 

119, A,B, and C join together to purchase flour. A 
contributed $1800; B twice as much as A, and C as much 
as A and B, less $900. How many barrels of flour can they 
purchase at $9 a barrel? 

120. If 17 cows are worth $816, what is one cow worth? 
How much are 48 cows worth? 

121. Gave $4224 for 32 acres of land. At the same 
rate, what is the cost of one acre? Of 24 acres? 

122. If 75 horses are worth $11100, how much, at the 
same rate, are 56 horses worth? 63 horses? 

123. If 18 acres of land are worth $3366, for how much 
per acre should it be sold so that $720 may be gained? 

124. Wow many tons of coal at $9 a ton will pay for 84 
thousand feet of lumber at $36 per thousand feet? 

_ 125. By selling 31 acres of land for $3100, I lose $155. 
What is the cost per acre? Cost of 16 acres? 

126. If 180216 square rods of land be divided equally 
among 12 men, what is the value of each man’s share at 
$1.25 per aS rod? 

127. 4 of $42075 is what a gentleman paid for a hence 
and lot, atk was $568 more than what he sold them for. 
How much was received for them? 

128. $36864 is 12 times what I paid for 96 acres of 
Jand. What is the cost per acre? Of 35 acres? 

129. Wow many days will 128200 pounds of flour last 
641 men, giving each man 4 pounds each day? 

150, A merchant sold 63 barrels of oil for $1008, gain- 
ing $126. How much did the oil cost per barrel? Cost of 
36 barrels? 

151. By selling 145 tons of hay for $2320, I lost $435. 
What did one ton cost? Cost of 48 tons? 

132. Bought 96 horses for $12000, and sold 78 of them 
at $136 each, and the rest at cost. How much did I gain? 


DIVISION. 91 


188. Sold 48 acres of land for $5520 and gained $480. 
What was the cost of 27 acres? 

134. Bought 42 carriages for $5376. For how much 
should I sell 18 of them so as to gain $126? 

135. JV£ 1 sell 124 head of cattle for $3720, and lose 
$620, for how much should I sell 56 head to gain $3727 

136. A,B,C, and D, wish to buy coal; A and B each 
furnish $2500; C furnishes as much as both A and B; and 
D furnishes as much as the other three. How much coal at 
$8 a ton can they buy? 

187. Vf $7560 be divided so that A shall receive $175 
more than 4 of it, B $1250 less than 4 of it, and C the 
remainder, how much will each one receive? 

138. A farmer sold a grocer 20 pounds of butter, at 18c. 
a pound; 17 dozen eggs, at 12c. a dozen; 9 bushels of pota- 
toes, at 60c. a bushel, and received in payment 54 pounds 
of sugar, at 14c. a pound, and the remainder in rice, at12c¢. 
a pound. How many pounds of rice did he get? 

139, A gentleman owned ,'5 of a tract of land of 12960 
acres, and divided it equally among his 8 children. What 
is the value of each child’s share, at $36 an acre? 

140. If the divisor is 341, and the quotient 589, what is 
the dividend? 

141, From the sum of 49 thousand 5, and 3 thousand 2, 
subtract their difference, and divide the remainder by 204. 

142. <A grain dealer delivered to a customer 3240 pounds 
of wheat. If there are 60 pounds in one bushel, how many 
bushels did he deliver? 

L435, If 48 bushels of oats weigh 2304 pounds, how 
many pounds will 75 bushels weigh? How many pounds 
will a car-load of 536 bushels weigh? 

144. A, B, and C, bought a farm for $9580; A con- 
tributed $125 less than + of the whole sum; B $1584 more 
than + of it, and C the remainder. How much did each 
one contribute? 


145. Bought a tract of land and divided it into 72 house 


92 HLEMENTARY ARITHMETIC. 


lots, which I sold at $248 each, thereby gaining $6912. 
What was the cost per lot? 

146. B, C, and D, purchased 264 acres of land for 
$11850; B’s share of it was 12 acres more than 4 of it; C’s 
share 24 acres more than + of it, and D’s the remainder. 
How much did each one pay? 

147. If two poems contain respectively 15693 and 9892 
lines, in how many days car¥ a boy read both of them if he 
reads 85 lines each day? 

148. A sold to a merchant 2460 pounds of wool at 45c. 
a pound, 1840 pounds at 36c. a pound; received $219.40 
cash, and the remainder in silk at $5 a yard. How many 
yards of silk did he receive? 

149. A fruit dealer sold 85 barrels of nuts, each con- 
taining 3 bushels, at $8 a bushel, and gained $255. What 
was the cost a barrel? 

150. <A farmer picked from one tree 320 apples; if he 
has 5 trees bearing the same number of apples, how many 
barrels can be filled from them, allowing 3 bushels to one 
barrel, and 40 apples to one peck? How many bushels will 
remain? 

151. Aman sold 64 calves at $7 each, and 29 sheep at 
$13 each, and with the money received bought 33 barrels 
of syrup. How much did he pay for four barrels? 

152. 87 cattle were found to weigh 34539 pounds. If 
they were of equal weight, how much more are 47 of 
them worth at 15c. a pound, than the remainder at 17c. a 
pound? 

158, Awiller earns $4.50 a day, and his expenses are 
$40 a month. How many months, omitting Sundays, will 
it take him to pay for a horse and wagon worth $1386, and 
a mortgage on his house of $476? 

154. A chaise wheel makes 349 revolutions in one mile. 
What is the distance traveled when the wheel has made 
17101 revolutions? What is the time consumed at 7 iniles 


an hour? 


DIVISION. 93 


155. A colony of 121 persons contribute $945 each 
towards the purchase of land. How many acres, at $21 an 
acre, can they buy? 

156. A brickmaker has 3 kilns, one of which contains 
36520 bricks, and each of the others 4 as many; after using 
7910 bricks, he sold the remainder in 43 loads. How many 
bricks are there in each load? 

157. Find the product of the sum and difference of four 
thousand ninety, and eight hundred seventy, and divide it 
by 905. 

158. Two drovers each have 954 sheep, costing $7 a 
a head; one of them sold them at a loss of $1 a head, and 
the other at a gain of $2 a head. How many sheep of an 
extra quality, valued at $30 a head, can they jointly buy 
with the proceeds of the sale? 

159. Wf the quotient is 17 when the divisor is 27, what 
will be the dividend if the quotient remain the same, and 
the divisor be doubled? 

160. There are 5280 feet in one mile. How often will 
a wheel 9 feet in circumference turn around in a distance 
of 18 miles? 

161. If acertain number be multiplied by 16, the pro- 
duct is 400; what is the product of the same number if 
multiplied by 192? 

162. Jf acertain number be divided by 125, the quo- 
tient is 48; what is the quotient if the same number is 
divided by 25? 

163. The divisor is 136, the quotient 98, and the re- 
mainder 105. What is the dividend? 

164. What number divided by 528 will give 36 for the 
quotient, and 44 for the remainder? 

165. What number multiplied by 86 will give the same 
product as 430 multiplied by 163? 

166. If 17 acres of land be sold at $28 an acre, and 
thereby $119 is lost, what is the cost of 29 acres at the same 
rate? 


94 ELEMENTARY ARITHMETIC. 


TO DIVIDE BY 10, 100, or 1000. 


Any integral number may be divided by 10, by 100, by 1000, etc., 
by pointing from the right of the dividend as many figures as there 
are ciphers in the divisor. 


Thus, 2846+ 10=284.6=284 and 6 tenths. 
2846-100 =28.46=28 and 46 hundredths. 
9846-1000 =2.846= 2 and 846 thousandths. 


1.—Divide 349 by 10. 7,—Divide 18469 by 100; by 
2.—Divide 867 by 100. 1000. 

3.—Divide 3468 by 100. 8.—-Divide 6984 by 100. 
4.—Divide 7962 by 1000. 9,.—Divide 1649 by 100; by 
5.—Divide 6459 by 100. ~ 1000, 

6.—Divide 8678 by 100. 10.—Divide 2564 by 1000. 


Removing the decimal point one order to the right multiplies the 
number by 10; removing the point two orders to the right multiplies 


by 100. 


Thus, 34.56 X10=845.6; 34.56 x 100=3456. 


Removing the decimal point one order to the left dzvzdes the number 
by 10; removing the point two orders to the left divides by 100. 


Thus, 678.9+-10=67.89; 678.9+-100=6.789. 


If a divisor ends in one or more ciphers, point from the right of 
the dividend as many figures as there are ciphers on the right of the 
divisor; divide what is left of the dividend by the significant fig- 
ures of the divisor. To the remainder, if any, annex the figures 
separated trom the right of the dividend. 


1.— Divide 643 by 90. . 


PROCESS. ANALYsIs.—Point the 3 from 

643-+90=64.3--9= 713 the right of the dividend, and di- 

+90=64.3+9=7T)§. vide 64 by 9. 64+9=7, and 1 

remainder. To this 1 annex the 

separated 3, making 18. The quotient is '7, with 18 remainder; or, 
the complete quotient is 713, 


2.—Divide 345 by 60. 6.—Divide 84692 by 2400. 
3.—Divide 8469 by 70. 7.—Divide 96728 by 3500. 
4.—Divide 3792 by 500. 8.—Divide 66729 by 4800. 


5.—Divide 7264 by 600. 9.—Divide 487629 by 23000. 


UNITED STATES MUNEY. 95 


SHENG PIU OWN Bea aie 


UNITED STATES MONEY. 


Art, 62. Currency is that which is given and taken 
as having value and representing property, and consists of 
coin or specie, and paper money. 

63. Coin or Specie is metal stamped, and authorized 
by law to be used as money. 

64. Paper Money consists of notes issued by the 
Treasury of the United States, or by banks, as substitute 
for coin. 

65. A Decimal Currency is a currency whose 
denominations increase and decrease in a tenfold ratio. 

66. United States Money is a decimal currency, 
and is often called Mederal Money; it was adopted by 
Cong-vess in the year 1786, as the currency of the United 
States. 

67. The coin of the United States consists of gold, 
silver, nickel, and bronze. 

68. The Gold coins are the one-dollar, three-dollar, 
quarter-eagle, half-eagle, and double-eagle pieces. 

69. The Silver coins are the trade-dollar, half-dollar, 
quarter-dollar, and ten-cent pieces. 

70. The Nickel coins are the three-cent and the five- 
cent pieces. 


71. The Bronze coins are the one-cent pieces. 


Notsr. — The silver five-cent 2nd three-cent pieces, and the bronze two-cent 
pieces, are no longer cvined. 


72. The gold coins are made of 9 parts pure gold and 


96 ELEMENTARY ARITHMETIC. 


1 part alloy, consisting of silver and copper. The silver 
coins are made of 9 parts pure silver and 1 part copper. 
The nickel coins are made of 75 parts copper and 25 parts 
nickel. The bronze coins are made of 95 parts copper and 
5 parts zine and tin. 


TABLE OF UNITED STATES MONEY. 


10 mills (m) - equal 1 cent, ¢ 

10 cents, - - ssh dime, «cr, 
10 dimes, or 100 cents, “ 1 dollar, $ 
10 dollars, - - “1 eagle, E. 


TABLE OF PARTS OF A DOLLAR. 


4 dollar=50 sents. 2 dollar =662 cents. 


4 dollar=334 cents. 3 dollar=75 cents. 
+ dollar=25 cents. 2 dollar=40 cents. 
4 dollar=20 cents. 3 dollar=60 cents. 
4 dollar=16% cents. 4 dollar=80 cents. 
4 dollar=124 cents. 2 dollar=374 cents. 
gy dollar=10 cents.  dollar=624 cents. 
qs dollar= 83 cents. & dollar =874 cents. 
ORAL. 


J. How many cents in one half a dollar? In a quarter 
of a dollar? 

2. How many cents in one eighth of a dollar? In ¢ of 
a dollar? 

&. How many cents in 3? of a dollar? In 2 of a dollar? 

4. How many cents in 4 of a dollar? In 3 of a dollar? 

5. What part of a dollar is 50 cents? 25 pee 

6. What part of a dollar is 75 cents? 125 cents? 374 
cents ?- 

7. What part of a dollar is oa cents? 874 cents? 334 
cents? 66% cents? 


UNITED STATES MONEY. 97 


S. How many cents in one tenth of a dollar? 3 tenths 
ot a dollar? 
9. How many cents in tof a dollar? Zof a dollar? 


NOTE.—The teacher should give thorough drill upon the aliquot parts of a dollar. 


73, The sign $ is written on the left of the figures 
which express dollars. 


Thus, $25 is read twenty-five dollars. 


¢4. Cents are hundredths of a dollar, and, when written 
decimally, occupy the two decimal places on the right of 
the decimal point, which separates cents from dollars. 
Thus, $9.25 is read nine dollars twenty-five cents. 


$6.08 is read s¢x dollars eight cents. 
$ .50 is read fifty cents. 


In these cases cents are regarded as hundredths of -the dollar, 
which is the unit. When the cent is regarded as the wn7t, no decimal 
point is necessary: as 10 cents, 125 cents, 5c., 8c., etc. 


3. Mills occupy the third decimal place on the right 
of the decimal point. 
Thus, $1.125 is read one dollar twelve cents five mills ; 


Or, one dollar twelve and one half cents. 
$ .005 is read five mills, or one half cent. 


WRITTEN. 


10. Add four dollars fifty cents; eleven dollars twenty- 
five cents; 30 dollars 40 cents; 7 dollars 13 cents. 

11. Add 8 mills; 20 cents; 1 dollar; 2 dollars 30 cents; 
twelve cents five mills. 

12. $54+83.75+$84.00+8.15 +$.125 + $1.05 -+$.008 =? 

13. $1.75 +$2.50+$3.25 +$1.125 +$.75 +8.375?= 

14. $10$+$24+48734 $234 94.00=? | 

15. $8443.75 +892 +4$3.25 +$8.625 48.875 =? 

16. %8400+8961+8185.00 +837.50 + $483.40 +8175 =? 

17. + $35+$35.00 +$3500 +$.35 +$3.50+%.035 =? 


98 ELEMENTARY ARITHMETIC. 


18. $184—$35=? 24. $846.59—$150=? 

19. $32.50—$28.75=? 25. $137.945—$84.30= 

20. $1340—$875.50=? 26. $12.75—81 875=? 

21. $9837.40—$1568. ea 27. $93.625—$854=? 

22. $235.92—$146.875= 28. $18465$—$387.25=? 

23. $1000—$48.375=? 29. $9643.80 —$46734=? 

2U.. tigip Xlo—e 36. $39.40xK124=? 

31. $1200K29=? 37. $324.60 374=? 

32. $83.75 x 34=? 38. $596.25 334=? 

Od» pic teet aed 39. $95.625 x 624=? 

34. $864.50X60= 40. $194.00 x 100=? 

oo. $12,375 X15=—? Al. $137.255 X 125= 

42. $1000+4=? 48. $3275.60+48=? 

43. . $3750+—25=? 49. $5973.64+124=? 

44, $9640+80=? 50. $375.00-+45= 

45. $58.50+48=? 51. $250.50+-50=? 

46. $1.875+24=? 52. $182.04+64= 

47. $5.625+48=? 53. $39.406+124=? 
BILLS. 


76. A Bill of Goods is a written statement of articles 
sold, with the date of sale, the price or value annexed to 
each article, and is given by the seller to the buyer. 

vi, An Account is a written statement of debits and 
credits in business, as between two parties, the buyer and 
the seller. : 

78. A Debit is a written record of what is due or 
expected from the buyer to the seller. 

79. A Debtor is the party from whom payment is due 
or expected. 

80. A Credit is a written record of payment, in part 
or in full, made by the debtor. 


UNITED STATES MONEY. 99 


81. A Creditor is the party to whom payment is 
due, or by whom it is expected. 

82. When payment is made in full of a bill, it is 
receipted, and is then called a receipted bill. 


NoTe.—The following are simple forms of bills, serving as models, to which the 
teachers should add daily from dictation until pupils are familiar with the business 
forms. 


ah Chicago, Jan. 1, 1876. 
James Watson, 
To Gro. SHERWOOD & Co., Dr. 


1875. 
oe 1 | To4 doz. Model First Readers, - @ $3.50 $14 | 00 
Nov. 10} “ 5doz. Model Second Readers, @ $4.00 || 20 | 00 
Nov. 30 | “ 12 doz. Writing Spellers, - - @$1.10|| 13 | 20 
Dec. 20! “ 12doz. No.2 High School Slates,@ $1.50 |! 18 | 00 $65 !20 


Received payment, 
GEO. SHERWOOD & Co. ° 
Per Sprague. 


2. Cincinnati, Jan. 25, 1876. 
Theodore Smith, 
Bot of Witson, Huxirt & Co. 


32 pieces of Brussels Carpet, 56 yards each, @ $3.80 . 
48 pieces of Wilton % 64 yards “ @ 3.25 . 
50 pieces of Turkey 63 yards “ @ 340. 
64 pieces of Ingrain “ 48 yards “ @ 1.75 


8. _ Columbus, O., April 1, 1876. 
Mr. George Caldwell, 


To Hiitit, Jounson & Co., Dr. 


1876. 
Jan. 1. To 36 yards of Linen, @_ $2.50 . 


Feb. 1. “ 40 yards of Cambric, @ 60c. 
March 1. “ 60 yardsof Clothh @ 800... 
March 15. “ 24 yards of Velvet, @ 16.00... 


100 ELEMENTARY ARITHMETIC. 


4 St. Louis, June 1, 1876. 
Mrs. James Shields, 


Bought of M111, Brown & Co. 
1876. 
April 15. 24 yards of Cloth, - @ $3.80... 
April 25. 13 yards of Velvet, @ 9.00. . 
May 10.. 36 yards of Silk, - @ 4.50. . 
May 25. 27 yards of Cashmere,@ 2.75. . . 


5. Springfield, Ill., July 1, 1876. 
Mr. Thomas Barnes, 
Bought of Stone, ANDERSON & Co. 


1876. 
May 28. 24 tons of Lackawanna Coal, @ $7.50. . . 


May 30. 30 tons of Briar Hill Coal, @ 7.00. . . 
June 12. 45 cords of Maple Wood, @ 8.00. . . 
June 18. 386 cords of Beech Wood, @ 7.50. . . 


2 Madison, Wis., June 25, 1876. 
Messrs. Paul, Everett & Co., 
To Wiiu1aAM NEtson, Dr. 


1876. ; 
Jan. 10. To 125 barrels of Flour, @ $7.80... 


Jan, ©1824 - “2386 bushels of) Wheat,) <@s-20; ao.ma 
Feb. 5. “ 190 bushels of Corn, Ct SOchsss 
- “ « 80 sacks of Potatoes, @ 1.80.-. . 
March 20. “ 48 sacks of Corn Meal, @ 2.75... . 


REVIEW QUESTIONS. 


What is the swm of several numbers? What is addition? What 
sign is used to indicate addition? What are addends? Describe the 
sign of equality? What is the principle applicable to addition of 
numbers? How should numbers be written for addition? At which 
order begin the operation® How may errors in addition be easily 
detected ? 


—~e 


UNITED STATES MONEY. 191 

What is meant by the d¢fference of two numbers? Define subtrac- 
tion. What terms are employed in subtraction? Define ménwend. 
Define subtrahend. What sign is used to indicate subtraction? When 
is the difference called remainder? What is the principle applicable 
to subtraction of numbers? How should numbers be written for 
subtraction? At which order begin the operation? Ifa figure in any 
order of the minuend is less than the figure in the same order of the 
subtrahend, how is the deficiency supplied? What is a proof of 
subtraction ? 

What is multiplication? By what other process may the same 
result be produced? What terms are used in multiplication? Define 
multiplicand. Define multiplier. Detine product. What sign is used 
to indicate multiplication? What are the multiplicand and multi- 
plier often called? What is the principle first named applicable to 
multiplication? Second principle? Third principle? Fourth prin- 
ciple? Of what denomination is the product? What is meant by 
partial product 2? What are the proofs of multiplication? How is the 
continued product of several numbers indicated? Mention two other 
methods of indicating the product of several numbers. 

How many forms of dévzston are there? Define first form. Define 
second form. What difference of written operation in the two forms ? 
What determines the particular form in oral and written analyses of 
examples? What are the terms used in division? Define dzv¢dend. 
Define divisor. Define quotient. Define remainder. How does the 
remainder compare with the divisor? What sign is ordinarily used 
to indicate division? In what other ways may division be indicated ? 
How is division related to multiplication? With reference to the 
dividend, what are the divisor and quotient called? What is the 
proof of division? What is the first principle applicable to division ? 
Second principle? Third principle? Fourth principle? In second 
form of division, how is the given part of the dividend found? At 
which order begin to divide? Of what denomination is the remain- 
der? What is meant by short division ? 

What is ewrrency? Of what does it consist? Describe cocn or 
specie. Define paper money. What is a decimal currency? Define 
United States money. By whom and when was it established as the 
U.S. currency? Of what does the coin of the U.S. consist? Name 
the gold coins. Name the silver coins. Name the nickel coins. 
Name the bronze coins. What is the composition of the gold coins? 
Silver coins? Nickel coins? Repeat the table of U.S. money. 

What is a bill of goods? An account? Define debit. A debtor. 
Define credit. A creditor. What is a receipted bill ? 


102 HLEMENTARY ARITHMETIC. 


SE. GLRON® Velde: 


PROPERTIES OF NUMBERS. 


FACTORS. 


The product of 3 times 4 is 12; 3 and 4 each is therefore called a 
factor of 12. Hence, 


Art. 83, A Factor of a number is one of the integers 
whose product is that number. 

It is also called a divisor or measure of the number. 

S4. The factors of a number may be unequal as the 
above, or they may be equal, if unity. be excluded, whose 
use does not affect the product. 

As, 2=2; the product of 2 times 2 is 4; the product of 2 times 2 
times 2 is 8. It will be seen that 2 is the only factor of 2, is one of 
the two equal factors of 4, and one of the three equal factors of 8. It 
is therefore called a root of 2,4 or 8. Hence, 

85. A Root of a number is the number itself, or one 
of the egual factors of the number. 

The root is designated as first, second, third, etc., root, according to 
the number of times it is used as a factor. 

86. The First Root of a number is the number 
itself. 

87. The Second Root of a number is each of the 
two equal factors of a number. 

The second root is usually called the Square Root. 

Thus, 3 is the square root of 9, since 9 is the product of 3X8. 

SS, The Third Root of a number is each of the 
three equal factors of a number. 

The third root is usually called the Cube Root. 

Thus, 2 is the cube root of 8, since 8 is the product of 2X2X2. 


—— —_— 


q 


PROPERTIES OF NUMBERS. 103 


89. There are numbers that have no factors but them- 
selves and unity. 

As, 3, whose factors are 3 and 1; 5, whose factors are 5 and 1; ete. 
Such numbers are called prime numbers. All other numbers are the 
product of factors each greater than unity; as, 15 is the product of 3 
and 5, 24 is the product of 4and 6. Such numbers are called com- 
posite numbers. Hence, 

90. Numbers, considered with respect of their factors, 
are prime or composite. 

91. A Prime Number is a number whose only 
factors are itself and unity. 

92. A Composite Number is the product of two 
or more integers, each greater than unity. 

93. A Prime Factor of a number is a factor which 
is a prime number. 


Thus, 2 and 3 are each prime factors of 6 or 12. 
94, A Composite Factor of a number is a factor 
which is a composite number. 


Thus, 4 and 6 are each composite factors of 12 or 24. 
Since 2 and 3 are each factors of 6 and 12, they are called com- 


mon factors of those numbers. Hence, 


95. A Common Factor of two or more numbers 
is a factor of each of the numbers. 

96. Numbers that have no common factor greater than 
unity are prime to each other. 


Thus, 4 and 5; 8 and 9 are prime to each other. 


97, One number is divisible by another, when the latter 
is a factor of the former. 


Thus, 3 being a factor of 6, 6 is divisible by 3. 


98. Even numbers are numbers which are divisible 
by 2. 
Thus, 2, 4, 6 and 8, are even numbers. 


99. Odd rumbers are numbers which are not divisible 
by 2. 


Thus, 3, 5,-7 and 9, are odd numbers. 


104 ELEMENTARY ARITHMETIC. 


100. Factoring is the method of obtaining the 
several factors or divisors of a number. 


PRINCIPLES. 


1. Every number is equal to the product of all its prime 
factors. 

2. A factor of a number is also a factor of any product 
of that number by integers. 

3. A common factor of several numbers is a factor of 
their sum. ° 

4. A common factor of two numbers is a factor of their 
difference. : 

101. To RESOLVE A NUMBER INTO ITS PRIME FACTORS. 

What are the prime factors of 60? 
PROCESS. 

2)60 


ANALysis.—By trial, 60 divided by the prime factor 2 
gives the composite factor 30 for a quotient; 30 divided 
2)30 by the prime factor 2 gives the composite factor 15 for a 
Legge quotient; 15 divided by the prime factor 3 gives the 
3)15 prime factor 5 for a quotient. Hence, 2, 2, 3, and 5, are 
the prime factors of 60. 


| 


Rule.—Divide the number by its least prime factor, and 
divide this and each succeeding quotient in the same manner 
until a quotient is obtained which is a prime number. The 
several divisors and the last quotient are the prime factors 
of the numbers. 

-What are the prime factors of the following numbers? 


Ve ads The? 21, 480. 
Q. 24. 12. 81. 22, 550. 
3. 27. 13: ooh 23. 585. 
4. 32. ly 96. Qh, 672. 
5. 42. 15. 108. 25. 512. 
6. 45. 16. 121. 26. 729. 
7, 49. ae 27. 156. 
8. 5A. 18. 240. 28. 860. 
9. 56. 19. 360. 29. 915. 
10. 63. 20. AS. 30. 1200. 


PROPERTIES OF NUMBERS. 105 


What prime factors are common to the following num- 
bers? 


31. 15 and 18. 36. 42 and 48. 
32. 18 and 24. 37. 42 and 49. 
33. 2Y and 36. 38. 56 and 72. 
34. 35 and 45d. oo. $8 and: 81. 
35. 36 and 45. 40, 72 and 84. 
CANCELLATION. 


102. Cancellation is the process of facilitating 
operations in division by rejecting common factors from 
both dividend and divisor. 

The Sign of Cancellation is an oblique mark drawn 
across the number in which is the canceled factor. 


Thus, 3, 3, 6, 9. 


1053. The operations in cancellation are dependent 
upon the following 


PRINCIPLES. 


1. The canceling of one of the factors of a number 
divides that number by the factor canceled. 
2. The canceling of one of a series of factors divides 
their product by the factor canceled. 
3. The canceling of equal factors from both dividend 
and divisor does not change the value of the quotient. 
1, Divide 3x68 x12 by 2x38x4x6 


PROCESS. Awatysts.— The factors of the dividend 
being arranged above, and the factors of the 
2xX8xBx 12 —19 divisor beneath a horizontal line, 3 is observed 
9xBx4x G =i, to be factor in both dividend and divisor, and 


is canceled from both, since by principle 3 the 
value of the quotient is not changed. 

In like manner, 6 is observed to be a factor of both dividend and 
divisor, and is canceled. 2, a factor common to 2 of the divisor and 
8 of the dividend, is canceled, leaving 4 the other factor of 8, in the 
dividend. 


A5* 


106 ELEMENTARY ARITHMETIC. 


4, a factor of both dividend and divisor, is canceled, leaving 12 of 
the dividend the only factor uncanceled, which is therefore the quo- 
tient. Hence the following 

Rule.—l. Write the factors of the dividend above and 
the factors of the divisor beneath a horizontal line. 

Il. Cancel equal factors of both dividend and divisor 
until the factors of the one are prime to the factors of the 
other. 

III. Divide the product of the remaining factors of the 
dividend by the product of the remaining factors of the 
divisor. 

Divide 4x6X8X10 by 2X3xK4~x5. 
Divide 5X7%9X11 by 3X5X7X11. 
Divide 6X8X12X18 by 4x6x9x 12. 
Divide 9X1215x18 by 6X5x9x12. 
Divide 10x14 21X28 by 3xX7xK4x14. 
Divide 15X20 X25 X27 by 10X15 x18 x 25. 
Divide 18 x 24 x 32X36 by 9x12x16x18. 
Divide 20x 3035 x40 by 10X15x7x«K8x5. 
Divide 24X28 x 36x45 by 16xX18x15x12. 
11. Divide 21x 2428x385 by 7K14xK 24x85. 
12. Divide 24x 30x 36x40 by 20x15x18 x12. 
18. (27X32 x 36 x 45)+(16 X9x12x«15)=? 
14. (28x 30x 35 x 40)+(14K 15k 20x 7)=? 
15. (80X35 x40 x 45)+(15 x 20 x 30 x 5) =? 
16. (32x36 x48 x54)+(16 X18 X 32 x 27) =? 
17. (86X42 x45 x 50)+ (18 x 21K 15 x 25) =? 

\+ 

)+ 


~ 
> 80 So NM Sd Sr bo 


18. (35x45 x50 x 55)+ (15 X 25% 35 XK 11)=? 

19, (40x48 x56 x 60) + (20 x 24 « 28 x 40) =? 

20. (45X63X 72 x 80)+(15 X21 x 24 x 80) =? 

21. A gentleman purchased 24 pounds of coffee, at 28c. 
a pound, and gave in payment 8 packages of sugar, each 
containing 7 pounds. What is the price per pound? 

22. Bought 12 car loads of coal, each containing 32 tons, 
worth $6 a ton, and gave in payment 8 boat loads of flour, 
each containing 36 barrels. What is the cost per barrel? 


PROPERTIES OF NUMBERS. 107 


23. A grocer exchanged 54 firkins of butter, each con- 
taining 48 pounds, at 25c. per pound, for 12 chests of tea, 
each containing 60 pounds. What is the price per pound 
of the tea? 

24, A company of wood-choppers cut 18 piles of wood 
of 36 cords each, and exchanged it for 12 car loads of coal 
of 42 tons each at $9 a ton. What is the price per cord of 
the wood? 

25. A farmer planted 3 fields of corn of 120 rows, each 
row containing 72 hills, and each hill 5 grains. How many 
fields could he have planted with the same corn, each field 
having 96 rows of 75 hills each, and each hill containing 6 


grains? 


COMMON DIVISORS. 


104. A Divisor of a number is any integer that 
will divide the number without a remainder. 

Thus, 5 is a divisor of 10 or 15. 

Since 10 and 15 are both divisible by 5, 5 is called a common 
divisor of 10 and 15. Hence, 

105. A Common Divisor of several numbers is 
a divisor of each of them. 

Since 5 is the greatest divisor of 10 and 15, it is called the greatest 
common divisor of those numbers. Hence, 

106, The Greatest Common Divisor of several 
numbers is the greatest divisor of each of them. 


ORAL. 


What integers are divisors of 6? 8? 10? 12? 
What integers are divisors of 9? 14? 15? 16? 
What numbers are divisors of 15? 24? 56? 72? 
What integers are divisors of 18 and 30? 24 and 36? 
What numbers are divisors of 49 and 56? 63 and 72? 
What numbers are divisors of 81 and 108? 84 and 96? 


DON So TN 


108 ELEMENTARY ARITHMETIC. 


WRITTEN. 
To FIND THE GREATEST COMMON DIVISOR OF SEVERAL NUMBERS. 
7. Find the greatest common divisor of 27, 36, and 45. 


PrRocEss, BY FACTORING. 
OOO KOO ANALysIs.—Separating the numbers into 
36=3x3x2x2 their prime factors, 1t 1s found that the 
La ar bees prime factors of 27 are 3, 3, and 38; the 
prime factors of 36 are 3, 8, 2, and 2; and 
the prime factors of 45 are 8,8, and 5. Upon inspection it appears 
tbat the only prime factors found in each of these numbers are 3 and 
3. Therefore, 3x5=9 is the greatest divisor of each of the numbers, 
and is therefore their greatest common divisor. Hence the following 
Ruatle,— Resolve the numbers into their prime factors, 


and find the product of the factors which are common. 


PRINCIPLES. 
1. A divisor of a number is a divisor of any product of 
that number by integers. 
2. The greatest common divisor of several numbers is 
either the least of the numbers or a factor of the least. 
Find the greatest common divisor of the following numbers: 


8. 24, 48, 64. 24. 280, 350, 420. 
9. 36, B4, 72. 25. 288, 396, 432. 
10. 21, 42, 84. 26. 384, 528, 576. 
11. 48, 64, 80. 27. 56, 648, 864. 
12. 45, 60, 75. 28. 16, 304, 380, and 456. 
18, 18, 36, 90. 29. 84, 336, 420, and 504. 
Lp. 24, 72, 96. 30. 92, 368, 460, and 582. 
15. 96, 112, 144. 31. 96, 384, 480, and 576. 
1G. 90, 126, 162. 32. 108, 432, 540, and 648. 
17. 60, 120, 180. 33. 600 ft., 700 ft., and 850 ft. 
oe 08) 11°72 126: 34. $720, $900, and $1080. 
oa =io2,- 144. 156. 36. 864c., 1080c., and 1296c. 
ZO Pela; 135, 180, 56. 672 in., 840 in., and 1008 in. 
21. 128, 160, 208. 37. 125 As 375 Aw and 500 A. 
22. 180, 240, 400. 38. $132, $396, $528, and $660. 


23. 270, 360, 540. 39. $144, $432, $576, and $720. 


PROPERTIES OF NUMBERS. 109 


MULTIPLES. 


The product of 4 times 5 is 20. The product 20 is therefore called 
a multiple of 4 or 5. Hence, 

107. A Multiple of a number is the product ob- 
tained by using that number as one of its factors. 

Factors and multiples are the reverse of each other, as in the above 
illustration, 4 is a factor of 20, and 20 is a multiple of 4. 

In the expressions 3=3, 3X3=9, and 3x3x3=27, it is observed 
that 3, 9, and 27 are respectively the products of one, two, or three 
factors, each equal to 3, and they .are therefore called powers of 3. 
Hence, 

108. A Power of a number is the number itself, or 
is the product of several factors, each equal to the given 
number. 

109. These powers are designated as first, second, third, 
etc., powers, according to the number of times the given 
number is used as a factor. 

110. The First Power of a number is the number 
itself. 

111, The Second Power of a number is the pro- 
duct of the number used twice as a factor. 


The second power is usually called the Square. 
Thus, 16 is the square of 4, since it is the product of 4x4. 


112. The Third Power of a number is the pro- 
duct of the number used three times as a factor. 

The third power is usually called the Cube. 

Thus, 27 is the cube of 3, since it is the product of 3X33. 

115. The Haponent of a power is a number written 
to the right, and a little above the number, and shows the 
number of times the number is used as a factor. 

Thus, in the expression 4°, which is read 4 sguare=4X4=16, 2 is 
the exponent, and shows that 4 is used twice as a factor. 4° which is 


read 4 cube, =4X4%4—64, and the exponent 3 shows that 4 is used 
as a factor 3 times. 


114, The method of obtaining powers, though iden- 


110 ELEMENTARY ARITHMETIC. 


tical with all operations of multiplication, is called Invo- 


lution. 
ORAL EXERCISES. 


Name a number of which 3 is a factor; 4; 5; 6; 8. 
Name a number of which 9 is a factor; 10; 12. 
What is a multiple of 7? 11? 20? 30? 25? 

Name a number of which 2 is an equal factor; 3; 4. 
Name a number of which d is an equal factor; 6; 7. 
Name a number of which 8 is an equal factor; 9; 10. 
What is the first power of 4? 6? 8? 12? 100? 
What is the square of 5? 7? 9? 11? 12? 

. What is the cube of 2? 3? 4? 5? 6? 

10. What is the value of 37? 42? 62? 82? 92? 

11. What is a factor of 12? 15? 24? 36? 40? 

12. -Name a factor of 21; 27; 32; 42; 46. 

13. What is an equal factor of 9? 16? 25? 36? 49? 
14, What is the first root of 7? 8? 15? 48? 72? 

15. What is the square root of 16? 49? 64? 81? 121? 
16. What is the cube root of 8? 27? 64? 125? 216? 
17. What is the value of 4? plus the square root of 25? 
18. What is the value of 53 plus the cube of 2? 


SESS SF SUS 86S 


COMMON MULTIPLES. 


Since 12 is a multiple of both 3 and 4, it is called a common 
multiple of those numbers. Hence, 


115. A Common Multiple of several numbers is 
a.multiple of each of them. 

Again, since there is no number less than 12 which is a multiple 
of both 3 and 4, it is therefore called the least common. multiple of 
those numbers. Hence, 

116. The Least Common Multiple of several 
numbers is the least number that is a multiple of each 
of them. 


PROPERTIES OF NUMBERS. 111 


117. To FIND THE L&East Common MULTIPLE OF 
SEVERAL NUMBERS. 


1, Find the least common multiple of 9, 10 and 12. 


PROCESS. ANALYsISs.—Since the num- 
9=3x<3 ber sought is a multiple of 
~ See re 2 i 
10=2x5 the numbers 9, 10 and 12, it 


Mtn ee should be large enough to 
12=2Xx2x3, contain all the factors, of each 
L. C. M. is 2X¥2%3K5X3=180, of them, and since it is the 

least multiple of these num- 
bers, it should be only large enough to contain the factors of each of 
them and no other factors. The least common multiple of 9, 10 and 
12, will contain 12, or all the factors of 12, which are 2, 2 and 3. 
Since it will contain 10, it will contain all the factors of 10, which are 
2 and 5; but the factor 2, found twice in 12, includes the factor 2 found 
once in 10: hence the 2 in 10 is rejected, and the other factor, 5 in 10, 
is retained as an additional factor of the least common multiple, so 
that it will contain all the factors of-10. Since the least common 
multiple will contain 9, it will contain all the factors of 9, which are 
3 and 3; but the factor 3 found in 12 includes one factor 3 found in 9; 
hence, one factor 3 in 9 is rejected, and the other factor 3 in 9 is 
retained as an additional factor of the least common multiple, so that 
it will contain all the factors of 9. Therefore, all the factors of the 
least common multiple are 2, 2, 3,5 and 3, and their product, which 
is 180, is the least common multiple. Hence the following 


— 


Rule.—I. Separate the numbers into their prime 
Jactors. 

Il. Find the product of all the factors of the largest 
number, and such factors of all other numbers as are not 
Sound in the largest number. 


PRINCIPLES. 


1. Every multiple of a number contains all its prime 
factors. 

2. A common multiple of several numbers contains ail 
the prime factors of each of them. 

3. The least common multiple of several numbers con- 
tains all their prime factors, and no others. 


112 ELEMENTARY ARITHMETIC. 


4. The least common multiple of several numbers con- 
tains each of their prime factors the greatest number of 
times that it appears in either number. 


Find the least common multiple of the following numbers. 


2. 12. 16. 24. 36. 16. 36. 72. 108. 144, 
Bi-16. 24.7 80.40. 17. 40. 80. 120. 160. 
1S Ore) Ree ap. 18." 45. 780.0 200.188 
5. 20. 25. 30. 35. 19. 48. 72. 96. 120. 
G. 24, 36. 48. 54. 20. 56. 84. 112. 140. 
W532! \.4.0. yoann, BO. D1 San Ri a 
BO 866 45.0 BA: 163: 22. 60. 90. 150. 180. 
9. 40. 50. 60. 5. 23. 64. 96. 128. 160. 
70. 48. 60. 72, 84. G4. 12. 96. 144. 168. 
17. 45. 60. 7. 90, 95. 42. 84. 108. 180. 
12, 36... 54. 72. 900s 26. 75. 100. 125. 160. 
TS cAD.“& 60/40 8Or 00: 27. 80. 120. 160. 200. 
14, 25. 50. 175. 100. 28. 96. 108. 120. 182. 
15. 30. 60. 90. 120. 29. 96. 120. 182. 156. 
PROBLEMS. 


30. What is the least distance that can be exactly 
measured by either a yard measure, a as 8 feet in length, 
or a pole 16 feet in length? 

31. What is the size of the smallest tract of land that 
can be divided into lots of either 10 acres, 12 acres, 15 acres, 
or 18 acres? How many lots will there be of each? 

32, What is the capacity of the smallest vessel that 
can be exactly filled by either an 8 gallon keg, a 21 gallon 
vessel, a 40 gallon cask, or a 63 gallon hogshead? 

33. What is the smallest sum of money that can be 
used to purchase sheep at $9 a head, hogs at $15 a head, 
cows at $45 a head, or horses at $90 a head? How many 
of each may be bought? 

34. What is the least sum of money that may be used 


PROPERTIES OF NUMBERS. Ls 


to buy land at $24 an acre, $60 an acre, wagons at $108 
each, or suburban lots at $240 each? 

30. A, B, C,and D, start together at a certain point to 
travel around a certain island. A can pass around it in 12 
hours; B in 15 hours; C in 18 hours, and D in 24 hours. 
In how many hours will they all be together again? How 
many times will each one pass around the fiand before 
coming together? 


REVIEW QUESTIONS. 


What is a factor of a number? By what other names is a factor 
called? Define root of a number. How is the root of a number 
designated? What is the first root of a number? The second root? 
By what name is the second root usually known? What is the third 
reot of a number? By what name is it usually known? How are 
numbers considered with reference to their factors? Define prime 
number. Define composite number. What is a prime factor of a 
number? Composite factor? What is a common factor of several 
numbers? What numbers are prime to each other? When is one 
number divisible by another? What are even numbers? Odd num- 
bers? Define factoring. What is the first principle applicable to 
factoring? Second principle? Third principle? Fourth principle? 
How is a number resolved into its prime factors ? 

What is cancellation? Name the first principle upon which oper- 
ations in cancellation are dependent. Second principle. Third 
principle, Give the rwile for cancellation. 

What is a dévésor of anumber? Define common divisor of several 
numbers. Define greatest common divisor of several numbers. Give 
the rule for finding the greatest common divisor of several numbers. 
What is the first principle applicable to the finding of the greatest 


common divisor? Second principle? 


Define multiple of a number. What is a power of a number? How 
are powers designated? Define first power of a number. Second 
power. By what name is the second power usually known? Define 
third power of a number. By what name is it usually known? 
Define exponent of a power. Define ¢nvolution. What is a common 
multiple of several numbers? Define least common multiple of several 
numbers. Give rule for finding the least common multiple. What 
is the first principle applicable to the finding of the least common 
multiple? Second principle? Third principle? Fourth principle? 


114 ELEMENTARY ARITHMETIC. 


obs bal Neral eee 


FRACTIONS. 


Art. 1218, When any thing is divided into two equal 
parts, each part is called one half. 

There are two pints in one quart; one pint is therefore one half of 
one quart. One half of two apples is one apple. One half of 100 
apples is 50 apples. 

When any thing is divided into three equal parts, each 
part is called one third. ‘Two of the parts are called two 
thirds. 

There are three feet in one yard. One foot is therefore one third 
of one yard; two feet are two thirds of one yard. 

There are two halves in one thing; three thirds; four 
fourths; five fifths; ete. 

119. A Fraction is one or more of the equal parts 
of a unit, or of any number regarded as a unit. 

If an orange is divided equally among six boys, the part which 
each boy receives is called a fraction. Two of the parts are also 
called a fraction; so are three of them, or more. 

120, The number or object which is divided into equal 
parts is called the Unit of the Fraction. 

Each of the equal parts into which the unit of the frac- 
tion is divided is called the Fractional Unit. 


In the illustration given above (Art. 119), the orange is the unit of 
the fraction; one sixth of the orange is the fractional untt. 


121, Fractions are represented by figures, as follows: 


. 


FRACTIONS. 115 


One half is written $. Two halves are written 3. 
One third r ay 4. Two thirds “ tk 
One fourth . ey Three fourths *“ oh mane 
One eighth - ot gp Seven eighths “ heey 
One sixtieth ‘“ - Forty sixtieths 40, 


122. The numbers, or Terms, which represent frac- 
tions, are called Neemerator and Denominator. 

7123, The numerator is written above a short horizontal 
or oblique line, and the denominator below it. 4, 3, 2%. 

124. The Denominator shows the number of 
equal parts into which the unit of the fraction is divided. 

125, The Numerator shows either the unit of the 
fraction, or the number of fractional units in the fraction. 

When one orange is divided into s¢z equal parts, each part is re- 
presented by the fraction 4, in which 6 is the denoménator, and shows 
into how many equal parts the orange has been divided. The nwmer- 
ator. 1 shows that one orange has been divided. 

The fraction = represents two of the sZx equal parts into which one 
orange has been divided; or, it represents one szxth part of two 
oranges. The pupil can readily see that.one sixth of two oranges is 
equal to two sixths of one orange. 


126. A Mixed Number is composed of an integer - 
and a fraction. 14, 254. 


Read 

1..4 6. 4. Ll. $). 16. 5%. 
2. 3 7. +4. ° 12. +8. Pe wal tes 
3d. € 8. +r. 13, 1235, 18. 244. 
4. $ 9. Ay. LL. Pts. 19. 1812. 
5. £ LO. 4. 15. 442. 20. 29045. 
Write in figures: 

21. Four tenths. 26. Sixteen fortieths. 

22, Six ninths. 27, Twenty five-hundredths. 
23. Ten twelfths. 28. Thirty-six eighty-fifths. 


24. Bight elevenths. 29, Three and three eighths. 
25. 'Two sevenths. 30. Sixteen and five ninths. 


116 ELEMENTARY ARITHMETIC. 


12%, If three oranges are each divided into six equal 
parts, or stxths, there are eighteen sixths, 18. The number 
of oranges in these eighteen sixths, 18, can nai found by 
dividing the numerator by the denominator, 18+6=3. So, 
24—24-6=4; 24=24+8=3; etc. 

The quotient nie from the division of the numerator 
py the denominator is the Value of the Fraction. 

128, Whenever the numerator equals the denominator, 
the value of the fraction is one. 4=1. 

Whenever the numerator exceeds the denominator, the 
value of the fraction is greater than one. 12=3. | 

Whenever the numerator is less than ihe denominator, 
the value of the fraction is less than one. +4 41s less than 1. 
14 is less than 1. 

129. A Proper Fraction is a fraction whose 
value is less than one. 4, 11. 

130. An Improper Fraction is a fraction whose 
value is equal to, or greater than, one. 4, $. 

REVIEW QUESTIONS. 


What is a fraction? What are the terms of a fraction? Define 
each term. What is a mixed number? How is the value of a frac- 
tion found? What is the difference between a proper fraction and an 
improper fraction? Define each. 


REDUCTION OF FRACTIONS. 


CASE I.—To reduce an integer or a mixed number to an improper 
fraction. 


ORAL. 


131. 1. How many half oranges in 3 oranges? 


SoLutTion.—In 3 oranges there are 3 times as many half oranges 
as in 1 orange. In one orange there are 2 half oranges; in 3 oranges 
there are three times 2 half oranges, or 6 half oranges. 


sv. 


. FRACTIONS. 117 


How many thirds of an orange in 3 oranges? 
How many fourths of an apple in 4 apples. 
How many fifths of a dollar in ten dollars? 
In 5 how many halves? Thirds? Fourths? 
In 8 how many fifths? Sixths? Ninths? 
Change 6 to 10ths. To 12ths. 

Reduce 9 to fourths. To sevenths. 

Reduce 3? to fourths. 


SS COND OrpP So & 


SoLUTION.— 3}=3+-}. In3 units there are 3 times as many fourths 
asin 1 unit. In one unit there are 4 fourths; in 3 units there are 3 
times 4 fourths, or 12 fourths. 12 fourths+3 fourths=1/. 


Rule.—find the product of the integer and the de- 
nominator. To this product add the numerator of the 
Fraction, if there be any. The result is the numerator of 
the required fraction. 


Nore.—It will be observed that the rule directs the reduction of the integer to 
the same denoniinator as the fraction, previous to their addition. Thisis in accord- 
ance with the principle (Art. 33) that like numbers only can be added. 


10. Reduce 14 to halves. 14. Reduce 8% to 6ths. 


11, Reduce 2% to 3rds. 15. Reduce 771, to 10ths. 

12. Reduce 54 to dths. 16. Reduce 94 to 9ths. 

15. Reduce 6% to 4ths. 17. Reduce 102 to 7ths. 
WRITTEN. 


Reduce to improper fractions: 


18. 29:4. 22. 89-8,. 26. 81914. 
19: 40;1,. 93. Vz. Q7. 92st. 

22 O A24 D A()_5_ 
20. o624. BY. 8424, 28. 7405 1 . 

21. 37B,. 25. 6983. 29. 837412. 
CASE II.—To reduce an improper fraction to an integer or a mixed 
number. 

ORAL. 


132. 1. In 24 sixths of an orange, how many oranges? 


SoLuTion. — There are six sixths of an orange in one orange. In 
24 sixths of an orange there are as many oranges as 6 sixths are con- 


118 ELEMENTARY ARITHMETIC. 


tained times in 24 sixths, which are 4 times. Hence, in 24 sixths of 
an orange there are 4 oranges. 


2. In 48 how many units? 


SoLtution. — There are 3 thirds in one unit. In 4,2 there are as 
many units as $ are contained times in 4,2, which are 44 times. 


Hence, 4+,2=44. 


Rule. — Divide the numerator of the fraction by its 
denominator 


Reduce to integers or mixed numbers: 


cere 7, 36. 11. 9 15. 82. 
4. 22 B30. 12 39 16. 32. 
5, 48, 9, 49, 18, 52. 17. $3 
6. 89, 10. 58 12 a8 18, 4b 
WRITTEN. 
19. 1.92. QB. Bad OY, Be, 
20. 242, Qh, 325, 28, 40.9. 
21. agB. QB. Aga. 29, 825. 
22, 45.0, 26, 813. S) SgO} aS. 


CASE III.— To reduce a fraction to higher or lower terms. 


‘133. In one orange there are four fourths; therefore, 
in one half of an orange there is one half of four fourths, 
or two fourths; that is, ;=%. In one orange there are ten 
tenths; therefore, in one half of an orange there is one half 
of ten tenths, or five tenths; that is, }—,°,. These fractions, 
5, %, and >, have the same value; each is equal to one half 
of the same unit. They are, therefore, called equivalent 
Sractions. 

134. Equivalent Fractions are fractions of 
different expression, but of the same value. 

135. The value of a fraction is the quotient resulting 
from the division of the numerator by the denominator. 


(Art. 127.) That is, 


FRACTIONS. 119 


The, NuMERATOR is the Div1DENp. 

The Denominator is the Drvisor. 

The VALUE OF THE FRACTION is the QUOTIENT. 

Therefore, the principles of division apply to the terms 
of a fraction, and both terms of a fraction may be multiplied 
or divided b7 the same number, without changing the value 
of the fraction. One half maybe changed to two fourths, 
or three sixths, or ten twentieths, by multiplying both terms 
of the fraction 4, by 2, by 3, or by 10. So, two fourths, 
three sixths, and ten twentieths, may be changed to one 
half, by dividing both terms by 2, by 3, or by 10. 


ORAL. 


J. Change 2 of an orange to 10ths. 
SoLutTion.—One fifth is equal to two tenths. Then 3 fifths are equal 


to 3 times 2 tenths, which are 6 tenths. Hence, 3=,6,. 


Rule.—Multiply or divide each term of the fraction by 
the number necessary to change the given denominator to the 
required denominator. 


2. Reduce + to 8ths. 6. Reduce 4 to 20ths. 
3. Reduce ? to 8ths. 7. Reduce 2 to 24ths. 
4. Reduce 4 to 14ths. S. Reduce ;° to 40ths. 
}. Reduce # to 24ths. 9. Reduce ¢ to 36ths. 
10. Reduce § to 4ths. 14. Reduce 58 to 3rds. 
11. Reduce ;'; to 3rds. 15. Reduce 42 to 4ths. 
12, Reduce 4; to 5dths. 16. Reduce 2° to 3rds. 
13. Reduce 44 to 6ths. 17. Reduce 2% to 8ths. 
WRITTEN. 
18. Reduce ;8; to 84ths. 22. Reduce 5°, to 3rds. 
19. Reduce ;% to 100ths. 23. Reduce 4° to 4ths. 
20. Reduce 4° to 240ths. 24. Reduce 32° to dths. 
21. Reduce 8 to 288ths. 24. Reduce 4% to dths. 


120 _ ELEMENTARY ARITHMETIC. 


CASE IV.—To reduce a fraction to lowest terms. 


136. A fraction is in its Lowest Terms when its 
terms are prime to each other. 3, 14. 


ORAL. 


1. Reduce 48 to lowest terms. 


ANALYsIS.—Since (Art. 185) the value of a fraction is not changed 
by the division of both terms by the same number, each term of the 
fraction $$ may be divided by any factor common to both; and this 
division continued until the terms of the resulting fraction are prime 
to each other. 


ine 184-279 ‘oma 
SoLuTion I.—Divide each term first by 2. 37G=j5 Then divide 
each term by 3. a8 3 Since 8 and 4 are prime to each other, the 


SotuTrion IJ.—Divide each term by the greatest common divisor 
18+6 3 


of the terms. The G. C. D. of 18 and 24 is 6. ew ters 

Rule I,— Divide each term by any factor common to 
both terms; so continue until the terms are prime to each 
other. 

Rule T1.—Divide both terms by their greatest common 
divisor. 

Reduce to lowest terms. 


Fae ee Bye yg 6. 3 48s Fe. 

3. tes 353 oF 7. $05 433 oe 

20. 18:5. ese ‘BBs etl EL IO Ss: 

4. SOE le 488 3 8. 44% 81% 144° 

5. 363 255 $e 9. #8 3 a 

WRITTEN. 

10. W23 Hess Teo 14. tos 345 Dee- 
11. 053 i203 Tio 15. $503 803 Tos 
12. 2853 fos Hes 16. x85; veo3 13's: 
138. #303 feos 78 17. #ss3 ess aes 


CASE V.—To reduce several fractions to equivalent fractions having 
a common denominator. 


13%, Fractions having equal denominators are said to 
havea Common Denominator. 


wy oe | 


FRACTIONS. 121 


ORAL. 4 


1. Change the fractions 4 and 4 to equivalent fractions 


having a common denominator. 

Sonution I—Reduce } to4ths. 4equals #7. The required fractions 
are a and + 

Son UTION Le _Reduce to 8ths. 4=4; 4=%. The required frac- 
tions are 4 and 3 

Barone it Redes to 12ths. $=7°55 }= 75. The required 
fractions are 785 and 7°3- 

It is evident that any common multiple of the denominators of the 
given fractions may be taken as the common denominator. 


2. Change »%; and 42 to equivalent fractions having a 


common denominator. 
SoLurton.—Reduce to lowest terms.  9’y 
multiple of 4 and 5, say 20. By case III, ¢ 
Reduce to common denominator. 
?, 4 and 4. 5. #% and 4. 7. #3 and 33. 
. 2 and &. 6. #%and 4. &. 4£and 3. 


CASE vI.— To reduce several fractions to equivalent fractions having 
the Least Common Denominator. 

138, 1. Reduce $ and } to equivalent fractions having 
the least common denominator. 

ANALYsIs.—Any multiple of the denominators 2 and 3 may be 
taken for a common denominator; but it is evident that the east common 
denominator of the fractions is the least common multiple of the de- 
nomuinators 

SoLution.—The least common multiple of the denominators, 2 
and 8,is6 By Case III, $=3; $=#. 

2 Reduce 2, 2, and 3, to equivalent fractions having 
the least common denominator. 

SoLutrion.—The least common multiple of the denominators 5, 6, 
and 8, is 120. 


=120- _ 120— 24 - 7—2 20— 48 
1 505 4=4 of =f g=% of 453=745)- 
eG. | —1. tote. 2047 f=) 5 120—100 
1=153;3 ¢=3 of 438 =f'55 $—G Of 149 =192. 
—i120- 1—1 120— 15 - 3—3 of 120— 4 
1=}555 8 4 of 120 P20 8 3 of 453-4 iO 


Rutle.—Find the least common multiple of the denom- 


inators. This is the least common denominator of the 


46 


122 ELEMENTARY ARITHMETIC. 


Fractions. Multiply each term of each fraction by the quo- 
tient of the least common denominator divided by the denom- 
inator of the fraction. 


Reduce to equivalent fractions having the least common 


denominator: 
ORAL. 
o. ¢and 3. 7. 4and +4. ii. Zand 3. 
4. % and %. S&. # and 55%. 12. = and +. 
5. and $. 9. % and 4. 13. % and 4. 
6. 2 and 2. 10. % and 4. LZ. $and 44. 
WRITTEN. 
15. $, 4 and 3. 20. 2%, % and &. 
9 2) 1 4 3 re 
17. §%, 2 and §. 22. 4, % and 58. 
1 6 4 4 cg 2 
f 2 
19. 4, x5 and zy. 24. Fy, % and $. 


REVIEW QUESTIONS. 


How may an integer be reduced toa fraction? Will such a fraction 
be proper or improper? Why? How may a mixed number be 
reduced to an improper fraction? How may an improper fraction be 
changed to an integer or mixed number? When will the result be an 
integer? When amixed number? How may a fraction. be reduced 
to lower terms? To higher? What are the lowest terms of a fraction ? 

Define equivalent fractions. To what terms in division do the terms 
of a fraction correspond? What principles of division apply to frac- 
tions? How are fractions reduced to lowest terms? When do fractions 
have a common denominator? How are fractions reduced to equiva- 
lent fractions having a common denominator? Name several pur- 
poses for which fractions are thus reduced. How are fractions reduced 
to equivalent fractions having the least common denominator ? 


ADDITION OF FRACTIONS. 


i mORATS 


139. 1. William has 4 of a dollar, and Henry has 3 of 
a dollar. How many fifths of a dollar have both? 


FRACTIONS. 138 


Sotutron.—They both have the sum of 4 fifths of a dollar and 3 
fifths of a dollar, which is + of a dollar, equal to 1 dollar and % of a 
dollar : 

2. Mary learned her spelling lesson in 3 of an hour, and 
her reading lesson in { of an hour. In what time did she 
learn both lessons? 

3. Find the sum of +5, 33, and {&. 

4. Mr. Adams bought % of a ton of coal, and Mr. Bates 
bought # of a ton of coal. How much coal did they both 
buy? 

ANALysis.—Since # and ? have not the same fractional unit, that 
is, since 4 is not equal to 4, the sum is neither 5 thérds nor 5 fourths.’ 
But $ and # may be chanced to equivalent fractions whose denomina- 
tors are 12, 24, 36, or < ay other multiple of 3 and 4. 

Se ee =, 3 2 = "53 +35 + #5 =1i = 1,5. They bought 
the sum of ¢ of 3 ton and } of a ton, which is 13’, tons. 

Rule.—Meduce the fractions to equivalent fractions 
having a common denominator; find the sum of the numer- 
ators; divide this sum by the common denominator. 


Note.—It is generally best to reduce the fractions to their least common 
denominator. , 


5. Find sum of 3 and 2 

6. Find sum of A and L 

7. Find sum of } and 3. 

NOTE.—Reduce % to 9ths. 

S. James spent # of a dollar on Monday, 4 of a dollar 
on Tuesday, and 2 of a dollar on Wednesday. How much 
did he spend in the three days? 


9. #44=? 18. $+44+4=? 
10. 44+3=? i4. §$+§+3=? 

Ei oat pigeas 15. 44344=? 
12. y+sh=? 16. 84+3+q5=? 


17. Find sum of te and 5,%,. 


which is 13. “Add this to the sum of the integers 2 and 5. 
18. 1$442=? 20. 83-+53= 
19. 54434,=? 91. 8k+4,=? 


124 ELEMENTARY ARITHMETIC. 


WRITTEN. 
22. 3,1+4=? 30. 1324103+52=? 
23. Jg+3-=? OL. 293472,+185=? 
24. $+7,=? Of, 923+4+38412=? 
25. 4$4+2=? 83. 4$4+1544+603=? 
26. 39+4+5—=? G4 2024119+4513=? 
27. 34+84+%¢=? 85. Iie +-94244,,=? 
28. 441449 =? 30. 033, +167; 88=? 
29, 44+3+43=? OV. 724344+54A,=? 


38. A horse costs $1473; the harness $653; the buggy 
* $2313. Find cost of all. 

3&9, Mr. C’s house is worth $52752; his furniture $1300; 
his barn $7843. What are they all worth? 

40, One field contains 75% acres; a second field has 604 
acres; a third, 109;4, acres. How many acres in all? 


SUBTRACTION OF FRACTIONS. 


ORAL. 


140. 1. Mr. D. owned 3 of a mill, and sold 3 of it; what 
part of the mill did he then own? 

SoLuTion.—He owned the difference between % and % of the mill, 
which is % of the mill, equal to + of the mill. 

2. Mr. E. owns 3% of a ship, and Mr. F. owns ; of it. 
How much more does Mr. E. own than Mr. F.? 

3. Find the difference between 12 and 4%. 

4. Find the difference between % and 3. 

DO ese to equivalent fractions having least common 

eT rae 


denominator. a as re so so 30" 


Rule.—Reduce the fractions to equivalent fractions 
having a common denominator; find the difference of the 
numerators,; divide this difference by the common denom- 
inator. 


5. §-3=? 9. §—2=7 
3 a a oe 
JI _42—9? 3 eet 
+6 a 5 fe gE 34—7,=? 
Nie SS port bs 12, 4-3=? 


FRACTIONS. 125 


15. Find difference of 8} and 54. 

14. Subtract 153 from 20. 

15. 8$-43=? 

Sotutron I—The 4 in the minuend is equal to }. # can not be 
subtracted from }- Take 1 of the 8 units; reduce to 4ths, and add to 
the #. 4++7=}. 7§—43=33. 

SoLtution II.— 8—43=—3}; 344-4=33. 


‘16. 54-12=? 18. 72—58=? 
17. 33—-23=3 19, 64—23=? 
WRITTEN. 

20. 17%-—-83=? 22, 323—214=? 


ADDITION AND SUBTRACTION. 


24. $4+%—-4=? 28) $432? 
25. $+8—qy=? 29, 14+24—37,=? 
26. §—-3+8=? $0. 38+54-2,,=? 
27. $—-t+74=? $1. 34+43-8=? 


32. Mr. B. having $50, paid one bill of $72, and another 
of $102. How much money had he remaining? 

33. Mr. C. spends 3 of his time at work, $ in study. 
What part of his time is devoted to other things? 

34, 2%+52 are how much less than 8,5,? 

3d. 40% is the sum of 28, 153, and what other number? 


MULTIPLICATION OF FRACTIONS. 


CASE I.—To multiply a fraction by an integer. 


141. The principle in multiplication, that the product 
is of the same denomination as the multiplicand, is true in 
fractions as well as in integers. 5 times 3 oranges are 15 
oranges; 5 times 3 sheep are 15 sheep; 5 times 3 eighths are 


126 ELEMENTARY ARITHMETIC. 


15 eighths. The denominator of the fraction is the denom- 
ination, or kind, of the multiplicand and product. 


ORAL. 
i. Multiply 2 by 5. 
SoLtutron.— 5 times 3 e¢ghths are 15 eighths,=1{. 


2, Multiply 3 by 4. 4. Multiply 3 by 3. 
3. Multiply 4 by 2. 5. Multiply 3, by 7 ’ 


6. Multiply 2 by 3. 

ANALysts.—By the method used in solving oe 1, #x3=15= 

=24. This method multiplies the fractional unit $; the product, 4°, 
tains 5 times as many fractional units, or parts, as the multipli- 
cand, %- But the same result may be secured by making each 
fractional unit 3 times as large, instead of taking 38 times as many of 
the same size. Thus, 3 times + is $; 3 times ¢ are 3. Therefore, 
instead of multiplying the numerator of the fraction by the integer, 
the denominator of the fraction may be divided by the integer, where 
the integer is a factor of the denominator. 


5 — — 
SoLution.— § X 3 =g7g=3 —24- 


Rule.—Multiply the numerator, or divide the denomi- 
nator, by the integer. 


7. Multiply 2 by 2. 10. Multiply 31, by 5. 
8. Multiply -%, by 4. 11. Multiply 38 by 7. 
9. Multiply 2 ‘pe 8. 12. Multiply 3, by 8. 


13, Find the cost of 8 pounds of butter, at 4 of a dollar 
per pound. 

14, How many bushels of grain will a horse eat in 12 
days, if he eat 2 of a bushel in one day? 

15. What is the cost of 10 yards of cloth, at § of a 
dollar per yard? 

16. Multiply 32 by 5. 

SoLution.— 5 times 3=15; 5 times $=4,2=34}; 15+34=18b. 


17. Multiply 24 by 3. 19, Multiply 53 by 6 
18. Multiply 63 by 2. 20. Multiply 8% by 7. 


FRACTIONS. 127 


WRITTEN. 
21. §X25=? 24. {yX29=? 
22. 44x21=? 25. $x40=? 
23. %X33=? 26. #X25=? 


27. 42x14=? 


SuGGEsTION.—Cancel factors common to the denominator and the 
integer, 


28. #x15=? $2. 88x15=? 
29. #4x21=? 83. 9%xX20=? 
90. 4$x22=? 84. 642% 338=? 
31. 42x100=? 35. B&X56=? 


36, Find cost of 24 acres of land at $124 per acre. 

37. What is the value of 32 bales of cotton at $462 per 
bale? 

38, Cost of 4 houses worth $3251,%, each? 

39, How many miles can a locomotive run in 9 hours, at 
the rate of 36% miles an hour? 

40. Cost of 12 horses, each valued at $2103? 


CASE Ii.—To multiply an integer by a fraction. 


ORAL. 


142, 1. Multiply 8 by. #. 

ANALysIs.—To multiply 8 by # is to find # of 8. One fourth of 8 
is obtained by dividing 8 by 4; 3 fourths=3 times 1 fourth. 

SontutTion.— ¢ of 8 is 2; # of 8=3 times 2,=6. 


Special Rule.—Take such a part of the integer as is 
indicated by the fractional unit; multiply this quotient by 
the numerator. 


2. Multiply 9 by 4. 5. Multiply 25 by 4. 
3. Multiply 12 by §. 6. Multiply 40 by é. 
4. Multiply 14 by 2. 7. Multiply 60 by 5. 


128 


8. 


ELEMENTARY ARITHMETIC. 


Multiply 4 by §. 


SoLution.— 4 of 4is 4; 3 of 4 are 2 times $=$=24. 


9. 
LG 
Jig ke 


165. 


Multiply 5 by 2. 12. Multiply 4 by 3. 
Multiply 6 by 2. 13. Multiply 10 by 2. 
Multiply 8 by 3. 14. Multiply 6 by 2. 


Multiply 4 by 2. 


Notrt.—When the integer is a factor o1 the denominator, proceed as in 
3 
example 6, Case I. 4xX$=5 gga. 


16. 
Ly 


Dimes 
Multiply 5 by 4. 18. Multiply 6 by 35. 
Multiply 8 by +. 19. Multiply 10 ry 33. 


Since either factor may be regarded as the multiplier, Cases I and 
II, are identical in operation. #X7=03, and 7x 4=53. 


Rule.—Multiply the numerator, or divide the denomi- 
nator, by the integer. 


20. 
21. 
22. 
23. 


24, 


What is the cost of 3 of a ton of coal, at $9 per ton? 
Find cost of § of a barrel of flour, worth $7 a barrel. 
Find 8 of 3 bushels. 
Find 4 of 5 oranges. 


‘Multiply 3 by 23. 


SoLuTion.— 2 times 3 is 6; $4 of 3 is 14; 6+14=%4. 


25. 
26. 
Ae 
28. 


29. 
30. 
31. 
Jf. 
BS. 
SL. 


Multiply 5 by 23. 

Find cost of 6 yards of cloth, at 33 dollars a yard. 
Find cost of 12 pounds sugar, at 83 cents per pound. 
Find 9 times 73. 


WRITTEN. 
Multiply 24 by 58. 35. Multiply 49 by 38. 
Multiply 32 by 94. 36. Multiply 310 by 8. 
Multiply 108 by 63. 37. Multiply 270 by 103. 
Multiply 144 by 874. 38. Multiply 200 by 64. 
Multiply 84 by 2. 39. Multiply 312 by 58. 


Multiply 158 by 4. 40. Multiply 840 by 64. 


FRACTIONS. 129 


41. Find value of 384 acres of land at $102 per acre. 

42, How many miles can a ship sail in 184 days, sailing 
179 miles in 1 day? 

43, Cost of 40% tons of hay, at $9 per ton? 


CASE II1I.—To multiply one fraction by another. 


ORAL. 


143, 1. What is4of % of an apple? 
Awnatysis.—As one half of 2 wnzts is 1 unit, so one half of 2 thirds 
is 1 third. 
2. What is + of 4 of an orange? 
Evidently one fourth of an orange; since one half equals two 
fourths, one half of one half is one half of two fourths, which is one 
fourth. 


3. Multiply $ by 4. 


Anatysis.—This is equivalent to: Find $ of }- 4 of ae qs5 
4 of 4 are 4 times zz, or 7453 4 of % are twice as much as #4 of 4, 
that is, are 3°. 


2x4__ 8 
SOLUTION.— 3X 4= : 


8x5 15° 


Rule.—Divide the product of the numerators by the 
product of the denominators. 


4. Multiply 3x¢4. 7. Multiply 4x3. 
5. Multiply 3x4. 8. Multiply $x. 
6. Multiply 3x2. 9. Multiply 2x. 


10. What part of an apple is 3 of + of it? 

11. What part of a journey is 2 of 4 of it? 

12. Find cost of 2 of a yard of ail at #of a dollar a 
yard. 

15. A man owning # of a ship, sold + of his share. What 
part of the ship did he sell? 

14. If a man earn § of a dollar in 1 hour, how much 


. can he earn in 3} of an hour? 


15. Multiply $x# 


130 


HLEMENTARY ARITHMETIC. 


SUGGESTIONS.—Cancel factors common to both numerator and de- 


nominator. 


of the denominators is a divisor. 
both dividend and divisor does not affect the quotient. 
21. 
2S. 
DY. 
25. 


16. Multiply ?x¢4. 
17. Multiply 3x2 
18. Multiply $x¢4. 
19, Multiply 3x4. 
20. Multiply 4x18 


26. Multiply 24 by & 


The product of the numerators is a dividend ; 


the product 


The rejection of factors common to 


(Art. 105). 


Multiply 4x 7. 
Multiply +5 x 335. 
Multiply 3354 
Multiply 4x2. 
Multiply $x. 


SueGEstion.—Reduce 24 to an improper fraction. 


27. Multiply 34 by 14. 


SuecEstion.—Reduce both factors to improper fractions. 


28. Multiply 12 by 34 
29. lb e) 2 by 12. 


WRITTEN. 


30. Multiply i b 
31. Multiply ~ b 
32. Multiply 44x 

33. Multiply 118 3D £38, 


38, 
BY. 
40. 


Use Cancellation. 


42, What part of an acre of land is 7 of ¢ of 4 


NoTE.—The word of is here equivalent to the sign x. 


o4. 
DO. 
36. 
OF. 


Multiply 4°; by 54. 
Multiply 38 by 8,3, 
Multiply 83 by 63. 
Multiply 9% by 22. 


Cost of 80% acres of land at $23 per acre? 

Cost of 1125 pounds sugar at 84 cents per pound? 
Cost of ae yards cloth at $33 per yard? 

41. §X3XqaxX4XR=? 


of 2 of it? 


Fractions such as the 


above are called compound fractions, to distinguish them from such fractions as 


Siiae 
4? 9? 


41 ete., which are called simple fractions. 


144. A Simple Fraction is a fraction whose 


terms are both integers. 


145. A Compound Fraction is a fraction of a 


fraction, 


FRACTIONS. 131 


43. Find t of $ of 44o0f 2. 
44. Find of 4, of § of 32. 

45.. Find } of +8; of % of 8 of 4. 
46. Simplify }x2?x4x 7% xX#. 
47. Simplify 2x 35x 14*2x +5 


DIVISION OF FRACTIONS. 
CASE I.—To divide a fraction by an integer. 


ORAL. 
146. 1. Divide & by 5. 


This is equivalent to: Find + of 3 
SoLution.—One fifth of 5 alghths 1 is 1 eighth. 


2. Divide 4 by 4. 5. Divide 5 by 3. 
3. Divide £ by 3. 6. Divide 1% by 5. 
4, Divide 42 by 4. 7, Divide 18 by 6. 


8. Divide & by 2. 
AnNatystis.—Following the method used in example 1, that is, 
+ 2% 

dividing the numerator of the fraction by the integer, + Qn 93 "4. 
This result is correct, but is not in the form of a simple fraction. 
Instead, therefore, of dividing the numerator by 2, thus taking one 
half as many fractional units, it is better to take the same number 
of fractional a Sa half as large. That is, multiply the denom- 
inator by 2: $+2=— aaa It is also evident, that since je is $ 
of 4, 75 is + of 23; that is, 3+ ~2=5,. 


Sonution.— $+2=g79=75- 
Rule.—Divide the numerator, or multiply the denom- 
inator, by the integer. 


NOTE.—Observe, that as division is the opposite of multiplication, the rule for 
dividing a fraction by an integer is the opposite of the rule for multiplying a 
fraction by an integer. 


9. Divide $ by 2. 13. Divide § by 3. 
10. Divide 3 by 3. 14. Divide § by 5. 
11. .Divide 4 by 2. 15. Divide 2 by 4. 


12. Divide 3 by 4. 16. Divide 4 by 5. 


1382 ELEMENTARY ARITHMETIC. 


17. If % of a melon be divided equally between 2 boys, 
what part of the melon will each boy have? 

18. If 3 of a melon be divided equally between 2 boys, 
what part of the melon will each boy have? 

19. Tf % of a gallon of wine be divided equally among 
6 men, ae much will each man have? 


20. Divide 122 by 3. 
SoLuTion.— 12+3=4; 2+3=4; 444=41. 
21. Divide 123 by 5. 


SoLtutTron I.—Reduce the mixed number, 12? to an improper frac. 
; “ORI ORI ae. cel ee Bl aed 
tion ; 123= 7. 4°48 00 20" 

Sotution II.—Take from the dividend 123, the greatest multiple 
of the divisor, 5, contained in it; 122—10+22. Divide each part of 
the dividend thus separated, and add the quotients; 10+5=2; 24+5= 


eed 
7 Ay ase —_--_ —11 pi | it. 
bpp 2+34="ht 


4x5 
22. Divide 3% by 2. 26. Divide 12% by 3. 
23. Divide 54 by 3. 27. Divide 142 by 6. 
Qh. Divide 102 by 4. 28. Divide 154 by 7. 
25. Divide 158 by 5. 29. Divide 202 by 8. 


30. 8% apples were divided equally among 3 girls. How 
many did each receive? 

31, 124 bushels of oats were divided equally among 5 
horses. How much did each horse receive? 

32. If 94 barrels of apples were divided equally among 
8 families, how many would each receive? __ 

33. Divide 10$ sacks of flour equally aniong 6 persons. 


WRITTEN. » 
34. Divide 144 by 12. 39. Divide 125% by 6 
3). Divide 4,08 by 9. 40. Divide 2408 by 8. 
36. Divide 85 by 10. 41. Divide 1062 by 12. 
37. Divide 3% by 12. 42. Divide 2093 by 10. 


38. Divide 45 by 8. 43, Divide 312;% by 11. 


FRACTIONS. 138 


44. Tf aman walk 392 miles in 9 hours, how many miles 
can he walk in 1 hour? 

45. If a locomotive can run 496; miles in 12 hours, 
how many miles can it run in | hour? 

46. If 11 acres of land cost $1253, what is the cost of 
one acre? 

47. If 9 oxen cost $3202, what is the cost of one ox? 


CASE I1I.—To divide an integer by a fraction. 


ORAL. 


147. 1. How many halves are there in 1? How many 
thirds? Fourths? 

2. How many fifths are there in 2? In 4? In 5? 

3. How may times is } contained in 1 unit? 

SoLution.—There are 5 fifths in 1 unit. 3+$=5+1=5. 

4. Wivide 4 by 1. 

Sotution.— 4=2,9; #,0°+4—20+1=20. 

5. Divide 4 by 3. 

Sotution I— 4=2,; 2° +2—20+3-68. 

Sotution II.—The same result may be obtained by multiplying 
the integer by the denominator, and dividing the product by the 


namerator, of the fraction; that is, by multiplying the integer by the 
fraction ¢nverted. 


4x 
Thus, 4+3—4x $= ~ 2,0 —6%. 


Rule.—Multiply the integer by the fraction inverted. 


6. Divide 3 by 4. 9. Divide 8 by 3. 
7. Divide 5 by 3. 10. Divide 6 by 3. 
8. Divide 4 by 2. 11, Divide 7 by +5. 


12. Divide 2 oranges among some girls, so that each 
girl shall have $ of an orange. cates many girls will there 
be? 

13. How many boys can have ? of an apple each, if 6 
apples are divided among them? 


134 ELEMENTARY ARITHMETIC. 


14. How many yards of ribbon, at 2 of a dollar per 
yard, can be bought for 3 dollars? 

15. How many hours will it take a boy to walk 8 miles, 
if he walk $ of a mile an hour? 

16. Dvd 4 by 24. 

SOLUTION Reena each number to halves, 4=$, 24=$; 
¥+$=8+5=13. 


Sotution II.—Reduce 24 to halves; 24=3- Multiply the integer 
= a pee 13 3 


4 by the fraction 3 inverted, that is, by. 4x2= 


rt has 
17. Divide 3 by 12. 19. Divide 8 by 42. 
18, “Divide 5 by. 3h 20. Divide 7 by 33. 
WRITTEN. 
21. Divide 8 by +5. 25. Divide 6 by 28. 
22, Divide 12 by 4. 26. Divide 12 by 34. 
238. Divide 10 by 4. 27. Divide 9 by 124. 
24. Divide 9 by $. 28. Divide 7 by 10%. 


29. How many bushels of oats, at 2 of a dollar a bushel, 
can be bought for $200? bh: 

30. How many yards of cloth, at $43 per yard, can be 
bought for $90? 


CASE III.—To divide a fraction by a fraction, 


ORAL. 


148. 1. How many fourths in $? How many sixths? 
2. How many times is ¢ contained in 4? 
3. How many times is 4 contained in $? 


4. Divide § by %. 


NoTer.—It is evident that as 3 units are contained in 6 wnits 2 times, so are3 
eighths contained in 6 eighths 2 times. 


}. Divide 2 by 2. 


Sotution I.—Reduce the fractions to equivalent fractions having 
the least common denominator (Art. 188). #=$5; 2=s%. Divide 
the numerator of the dividend by the numerator of the divisor: 


15+-8=1f. 


FRACTIONS. 135 


Sotution II.—The same result may be obtained by multiplying 
the dividend by the divisor inverted. 


Thus, $+3=2X$=1f—=11. 


Rule 1,—Reduce the fractions to equivalent fractions 
having a common denominator. Divide the numerator of 
the dividend by the numerator of the divisor. 

Rule I1.—Multiply the dividend by the divisor in- 


verted, 
6. Divide 2 by 4. 10. Divide 2 by 4. 
7. Divide 4 by #. 11. Divide é by #. 
&. Divide 4 by 3%. 12. Divide ? by $. 
9. Divide # by 3. 13. Divide 2 by 3. 
14. Divide $ by 14. 
SOLUTION. Bane 13 to an improper fraction; 14= a 
pgs 


15. Divide 13 by 33. 


So.utrion.—Reduce to improper fractions; 13=3; 23=47. $+ 


33 
16. Divide 3 by 14. 18. Divide 24 by 34. 
17. Divide % by 24. 19. Divide 34 by 13. 


20. Into how many pieces, each 3 of an inch long, may 
a stick 25 inches long be cut? 

21. How many times may. a cup holding $ of a pint be 
filled from a jar holding 53 pints? 

22. How many pencils, costing # of a cent each, may be 
bought for 62 cents? 

23. Into how many balls, each weighing 14 ounces, may 
a piece of lead weighing 102 ounces be divided? 


WRITTEN. 
24. Divide {% by 3. 28. Divide 128 by 2,3. 
25. Divide 5 by }. 29. Divide 928 by 124. 


26. Divide 3% by 64. 30. Divide 324 by 64%. 
27. Divide 54 by 83. 31. Divide 342 by 818. 


136 ELEMENTARY ARITHMETIC. 


32. How many bottles, each containing 24+ pints, may be 
filled from a vessel containing 2043 pints? 

33. How many penholders, worth 23 cents each, may 
be bought for 50? cents? 

34, How many dozen lead pencils, worth $14 per dozen, 
can be bought for $243? | 


REVIEW QUESTIONS. 


How are fractions having a common denominator added or suh- 
tracted? What must be done with fractions which do not have a 
common denominator before they can be added or subtracted? What 
two methods of multiplying a fraction by an integer? When is the 
first method used? The second? State two methods of multiplying 
an integer by a fraction. When is the first method used? The second ? 
How is one fraction multiplied by another fraction? How is cancel- 
lation applied to this process? Why? How is a mixed number 
multiplied by a fraction or a mixcd number? How is a fraction 
divided by an integer? When is one method used? When the other ? 
How is an integer divided by a fraction? How is a fraction divided 
by a fraction? How is a mixed number divided by a fraction or 
mixed number ? 


MISCELLANEOUS PROBLEMS. 


ORAL. 


Change 34 to halves. 

Change 5% to 8ths. 

Reduce 74 to 7ths. 

Reduce 48 to integers. 

Reduce 8° to mixed number. 

Change $9 to whole or mixed number. 
Reduce 2 to 28ths. 

Reduce 3°, to 10ths. 

Reduce 2 and 3 to 36ths. 

Reduce 32 to an improper fraction. 


> SO So NSS St Ss ON 


— 


FRACTIONS. 187 


11. Reduce to lowest terms 32, ;%), 22. 

12. Reduce to lowest terms 4%, $8, =8)5. 

13, Reduce to common denominator 4 and 2. 

14, Reduce to least common denominator # and &. 
15. Reduce to least common denominator 3, # and 3. 
16. Add % and 3. 

17. Add %, $ and +. 

18, Find difference of # and ;%4. 

<9, Multiply # by 4. 

20. Divide # by #. 


21. 341s 4 of what number? 

SoLution.— 34 is 4 of 6 times 34, which is 20. 

22. 24is+ of what number? 

23, 124 is 4 of what number? 

24, 64 is 5 of what number? 

25. 8 is 2 of what number? 

SoLutron.—Since 8 is 3 of some number, 4 of that number is $ 


of 8, which is 4; and 3, or the whole number, is 5 times 4, which 
is 20. 


26. 9 is # of what number? 

27. 12 is 2 of what number? 

28. $18 is 3% of the price of a shawl. What is the 
price of the shawl? 

29, John, who is 15 years old, is $ the age of Henry. 
How old is Henry? 

30. If 4 of a ton of coal is worth $8, what is the price 
of 1 ton? 


31, After spending 4 of her money, Mary had 24 cents 
left. How many cents had she at first? 

SoLurron.—Since she spent + of her money, she had # of it left. 
% of her money equals 24 cents; 4 of her money is 4 of 24 cents, or 
12 cents; 4, or the whole, is 8 times 12 cents, which is 36 cents. 

32, James lost 2 of his marbles, and then had 30 remain- 
ing. How many had he at first? 


138 ELEMENTARY ARITHMETIC. 


33. Mr. D. sold 2 of his farm, and then had 500 acres. 
How many acres had he before he sold? 

o4. 4+ of a school was dismissed at recess, when there 
were 42 pupils remaining. How many pupils belonged to 
the school? 

35. If $ of a yard of cloth is worth $ of a dollar, what 
is the value of 1 yard? 

SOLUTION. = dince 3 $ of a yard is ae 5 of a dollar 4 of a yard 
is worth + of 3 of a dollar, ee is zo of a dollar, and $ of a yard, 
or 1 yard, is worth 6 times 75 of a alhte which is 99 of a dollar. 

36. If 2 of a sack of flour is worth $ of a dollar, what 
is 1 sack of flour worth? 

37. If 4 of John’s money equals ? of William’s money, 
what part of William’s money is John’s? 

38. #is % of what number? 

39, 341s 4 of what number? 

40. 4% is 4 of what number? 


41, % is the product of 4 and of what other number? 

42, 54 is the product of 7 and of what other number? 

45. The divisor is %, the quotient is 3. What is the 
dividend? 

44. The multiplicand is 24, the product is $. What is 
the multiplier? 

45. The multiplicand is 4, the multiplier is 2. What is 
the product? 

46, 2% is the product of 3 and of what other number? 

Pare a is the sum of what two equal numbers? 

48. TE is the sum of 4 and of what other number? 

49. 41s the niece tie of 14. and of what other number? 

50. What three equal numbers, when added, will pro- 
duce 1? 


51. 4 of a certain number exceeds 4 of it by 6. What 
is the number? 


SoLution.— The difference of } and § is 7p. 6 is 7’5 of the 


FRACTIONS. 139 


required number. 75 of the number is $ of 6, which is 2; the num- 
ber is 10 times 2, or 20. 

52. The difference between % and $ of a number is 5. 
What is the number? 

58. $10 is the difference between ? and 4 of the cost of 
a horse. What was the cost of the horse? 

54. Mr. E. owns } of a ship, and Mr. F. owns 4 of the 
same ship; Mr. G. owns the remainder. What part of the 
ship does Mr. G. own? 

55. Mr. H. owned % of a mill, but sold 2 of his share. 
What part of the mill did he sell? 

56. Mrs. K. had 4 of a gallon of syrup, and gave her 
neighbor 2 of what she had. What part of a gallon did 
she give? 

O7. Mrs. M. had 2 of a gallon of syrup, and gave her 
neighbor 2 of a gallon. What part of a gallon did she 
keep? 

OS. If 2 of a ton of hay cost 6 dollars, what will § of a 
ton cost? 

Suacestions.—Find the cost of + of a ton: then of 1 ton; then of 
tof a ton ; then of ¢ of a ton. 

O9, Vf % of an acre of land cost 25 dollars, what will 14 
acres nonne 

60. How many barrels of flour, at $54 per barrel, can be 
bought for $33? 

WRITTEN. 

61. Reduce 3215 to 9ths. 

62. Reduce ve to integer or mixed number. 

63. TE is dy of what number? 

G4. 203 is 2 of what number? 

65. Find sum of 24, 5%, and 84. 

66. Find difference of 82 and 5,8, 

67. Reduce to lowest terms 14% aie 60. 

68, What number is 34 less than 5%? 

69, What number is 23 more than 8}? 

70. What is ~ of § of 4 of 4? 


140 ELEMENTARY ARITHMETIC. 


71. The multiplier is 24, the multiplicand is 83. Find 
the product. 

72. The divisor is 32, the quotient is §. What is the 
dividend? 

73. Mr. A. had a farm of 640 acres. He sold 2 of it for 
520 per acre. What did he receive? 

74. Kind cost of 200 barrels of flour at $64 per barrel. 

7o. A man bought + of # of 400 acres of land. How 
many acres did he buy? 

76. Aman lost $233, which was $ of his money. How 
much had he at first? 

77. How many bushels of wheat, worth $1¢ per bushel, 
may be exchanged for 175 bushels corn, worth % of a dollar 
per bushel? 

78. If 2 of a yard of cloth cost 42 dollars, what will 33 
yards cost? 

79. At % of a dollar a pound, how many pounds of 
coffee can be bought for $32? 

SO. Find cost of 3% pounds of tea, at $14 per pound. 

SI. Mr. A. bought 5 houses; the first cost $3475.75; the 
second $2107.50; the third $4500; the fourth $1787.875; 
the fifth $2000. Find the average cost of each. 

S82. 2 of 8100 is what part of 3375? 

S83. Mr. D. lost $400, and had 4 of his money remaining. 
How much did he have at first? 

S4. If % of a vessel is worth $27000, what is the value 
of # of the vessel? 

S5. A number increased by 2 of itself is 28; what is the 
number? 

S6. The product of 3 numbers is 293; one of the num- 
bers is 54; another is 95. What is the third number? 

57. 3 of a farm is 48 acres less than % of it. How many 
acres are in the farm? 

SS. Mr. C. has 2 of his property in city real estate, ¢ of 
it in farming lands, and the remainder, $5700, in bank stock. 
What is he worth? 


DECIMAL FRACTIONS. 141 


oe Od i OO AN pee 


DECIMAL FRACTIONS. 


Art. 149, When a unit of any kind is divided into 10 
equal parts, each part is called one tenth. When one tenth 
is divided into ten equal parts, each part is one tenth of one 
tenth, and is called one hundredth. 

When one hundredth is divided into ten equal parts, each 
part is one tenth of one hundredth, and is called one thou- 
sandth. 

These fractions, +45, z45, qaly7, etc., increase and decrease 
in the ratio of 10; that is, each is one tenth of the next 
higher in the scale, and ten times the next lower. They are 
therefore called decimal fractions, or, more briefly, decimals. 

Observe that they increase and decrease exactly as inte- 
gers increase and decrease. Integers are decimals, for the 
same reason that decimal fractions are decimals; they in- 
crease and decrease in the ratio of 10. 

150. The denominator of a decimal fraction is always 
1, with ciphers annexed; that is, it is always a power of 10. 
(Art. 108). It is never 20, nor 50, nor 180; but it is always 
10, 100, 1000, 10000, ete. 

Thus, 75 is a decimal, but + and 4°, equivalents of 45, are not 
decimals, because their denominators, 2 and 20, are not powers of 10. 

It is not necessary to write the denominators of decimal 
fractions. °,; may be written .3; the period, here called the 
decimal point, distinguishes 3 tenths from 3 units. 3= 3; 
3.=3 units. 


142 ELEMENTARY ARITHMETIC. 


~3y may be written .03; i3’sy May be written .003; 
#55 may be written .25; q°5 may be written .125. 


Observe that when the denominator of a decimal fraction 
is not written, two things are necessary: Ist. The decimal 
point should be prefixed to the numerator of the fraction; 
2nd. The decimal must have as many figures as there are 
0’s in the written denominator. If there are not as many 
figures in the numerator as there are 0’s in the denomina- 
tor, 0’s must be prefixed to the numerator, when the denom- 
inator is not written. 

Thus, 5 =.9; qty =-18; tioa =-004; etc. 

151. The decimal point is always written at the left 
of the decimal fraction. It is sometimes called the separa- 
trix, because it separates decimal fractions from integers. 

Thus, .25=7%%53 but 2.5=23%;- The number on the left of the 
decimal point is an integer. 


152. A Decimal Fraction is a fraction whose 
denominator is a power of 10. 


ow 

gy TABLE. = 

3 mS 

SG 8 a oe 

° q aS é nn q g 
a lat Ss ects = ‘ 
A a ore ae Ry ae eae 
Sele oe fa Owes ee ~~ Me ishmeryt a ee 
= an ee eee ed S Sgt be, (Se See ae 
A le eet te ae ieee ae: aka ie ee 
Ge gO. EAA es Ae eee oe 
SEA e ee Piast er eee ou nciae spr iish ice 
ea OeaA Tae pR HH Ee Ba Ye se 
Lia at D ape 1 tae ee eee ie 
INTEGERS. DEcIMAL FRACTIONS. 

READING DECIMALS. 
1. Read .32. 


Sotutrion I.—The 3 occupies the place of tenths; the 2 the place 
of hundredths. 3 tenths and 2 hundredths are 32 hundredths. 

Sotution II.—Since the denominator of a decimal fraction is 
always 1 with as many 0’s as there are decimal figures, .32 must 
equal 7375, or 82 hundredths 


DECIMAL FRACTIONS. 148 


Rule.— or the numerator, read the decimal as though 
it were an integer. or the denominator, read the denom- 
ination of the right hand figure. 


Read the following: 


2. 8; 08; .008; .0008. 
8. .853y 48; .408;: 567. 
4. 896; 4564; 39275. 
5. 3.4; 3.43; 89.6756. 
6. .0908; 35.0006; 4.50004. 


WRITING DECIMALS. 


158. 7. Write 8 thousandths. 


SoLutTion.—Since thousandths is the name of the third decimal 
order, three orders of decimals are necessary in writing thousandths. 
Write the 8, prefixing two 0’s and the decimal point, thus, .008. 


&. Write 308 millionths. 


SoLution.—Since mdilionths is the name of the sixth decimal 
order, six orders of decimals are necessary in writing millionths. 
Write the 308 as an integer, and prefix three 0’s and the decimal point, 
thus, .000308. 


Rule.— Write the numerator as an integer. Prefia 0’s 
to the numerator, if necessary, until the number of figures 
in the decimal equals the number of 0's in the denominator. 
Write the decimal point at the left of the decimal. 


9. Write 29 ten-thousandths. 

10. Write 496 thousandths. 

411. Write fifty-six hundredths. 

12. Write eighty-four millionths. 

13. Write 166 ten-thousandths. 

14, Write 48 integers, and 64 hundredths. 

15. Write 896, and 35 thousandths. 

IG. Write 9846, and 29 millionths. 

17. Write thirty-four, and three hundred eight ten- 
thousandths. : 


144 ELEMENTARY ARITHMETIC. 


18. Write 46 tenths. 
Sotution.—46 tenths=44—4;5,=4.6. 


19. Write 124 hundredths. 
20. Write 4986 thousandths. 


154. By annexing 0 to a decimal, both terms of the 
fraction are really multiplied by 10, and the value of the 
decimal is not changed. 


Thus, .5=.50; that is, @>=7%%3 ete. 
PRINCIPLE. 


1. Annexing 0’s to a decimal does not alter its value. 


155. By prefixing 0 to a decimal, the denominator 
is multiplied by 10. The value of the fraction is there- 
fore divided by 10. 


Thus, 5=777; but .05=737; and zo is one tenth of 3r- 
PRINCIPLE. 

2. Prefixing 0’s to a decimal divides the decimal by 10 - 

as many times as there are 0’s prefixed. 


21. Change .4 to hundredths. 
22, Change .56 to millionths. 
23. Divide .8 by 10. 

24, Divide .34 by 100. 

25. Divide .49 by 1000. 

26. Divide .456 by 100. 


REDUCTION OF DECIMALS. 


CASE I.—To reduce a decimal fraction to a common fraction. 


156. 1. Reduce .8 to a common fraction. 
SoLution.— 8=75=4- 

2. Reduce .08 to a common fraction. 
Sotution.— .08=yt0=25" 


DECIMAL FRACTIONS. 145 


Rule.—Lrase the decimal point. Write the denomin- 
ator, and reduce to lowest terms. 


Nore.—When the decimal point is erased, the 0's at the left of a decimal become 
valueless, and should be omitted. Thus, 08=8. But.08 does not equal 8. 


Reduce to common fractions: 


Les 8. .96. 13. .800. 
ay-8. 9. 75. 14. 125. 
The HO AG 15. 375. 
6. 25. 11. .B5. 16. 875. 
7, A8. 12. 40. 17. .625. 


CASE II.—To reduce a common fraction to a decimal fraction. 


157. J. Reduce 3 to a decimal. 


SoLuTIon.— #=+ of 3. 38=80 tenths. + of 30 tenths is 7 tenths, 
with a remainder of 2 tenths. 2 tenths=20 hundredths. + of 20 
hundredths is 5 hundredths. Therefore, + of 8, or 4,=7 tenths-+-5 
hundredths, or 75 hundredths, or .75. 

2. Reduce 1; to a decimal. 


SoLution.— 1=10 tenths. 7g is 7g of 10 tenths, which is 0 tenth, 
with a remainder of 10 tenths; 10 tenths=100 hundredths; 3'¢ of 100 
hundredths is 6 hundredths, with a remainder of 4 hundredths; 4 
hundredths=40 thousandths; 3/¢ of 40 thousandths is 2 thousandths, 
with a remainder of 8 thousandths; 8 thousandths=80 ten-thousandths ; 
zg of 80 ten-thousandths is 5 ten-thousandths. 0 tenths +6 hundredths 
+2 thousandths+-5 ten-thousandths=.0625. That is, =4s=.0625. 


Rule.—Annex 0's to the numerator. Divide this result 
by the denominator. Place the decimal point at the left of 
the quotient. IPf necessary, prefix ciphers to the quotient, in 
order that there may be as many decimal orders in the 
quotient as there have been 0's annexed to the numerator. 


Reduce to decimal fractions: 


}. re ly 
g. Sait: 12. 
8, eet 18. 
he 10m ET oe Leake, 


ts So 


> Or 
pry 


146 RiLEMENTARY ARITHMETIC. 


15. Reduce 4 to a decimal. 


Sonution.--Anvex 0’s and divide by the denominator, according 
to the rule. Proceed thus as far as may be desirable, and annex the 
sign +. Thus, 3==.333-++. | 


Reduce to decimals of 4 figures each: 
16. 18. 
47. LI. 


es 
oh te 


REVIEW QUESTIONS. 


What is a decimal fraction? Are the denominators of decimal 
fractions usualiy written? Why is it not necessary to write them ? 
What is the decimal point? Why is it sometimes called the separa- 
trix? Repeat the rule for reading decimals. Rule for writing decimals, 
What are the two principles of decimals given? How may a decimal 
fraction be reduced to a common fraction? How may a common 
fraction be reduced to a decimal fraction ? 


ADDITION AND SUBTRACTION OF DECIMAL 
FRACTIONS. 


ORAL. 


158. d, -Add..6.and <9. 

Soxutron.---The sum of 6 tenths and 9 tenths is 15 tenths, equal to 
1 and 5 tenths, or 1.5. 

2. Add .7 and .4. 

&,. Add .21 and .12. 

4. Add .3 and .05. 

SoLution.— .8=.30; .380-+.05=.35. 

5. Add .5 and .09. 

6. Add .12 and .014. 


#59. The principles of addition and subtraction of | 
decimal fractions and of integers are the same, because both 
are decimal; that is, 10 units of any order are equal to 1 


DECIMAL FRACTIONS. 147 


unit of the next higher order. When numbers are to be 
added or subtracted, figures of the same order should be 
written in the same column; that is, units should be writ- 
ten under units, tenths under tenths, hundredths under 
hundredths, ete. All the decimal points will, of course, be 
in column. 


WRITTEN. 


7. Add 8.5, 25.8395, 69., 4.37, and .075. 


PROCESS. 
ae ‘ SoLution.—Write the numbers so that the decimal 
09.099 2 points shall all be in the same column. This will place 
69. figures of the same order in column. 
pe ¥ Add as integers, writing the decimal point between 
ale the integer and the decimal fraction in the result. 
107.7845 


&. From 237.5 subtract 68.625. 


PROCESS. So_ution.— Write the numbers so that figures of the 
237 500 same order shall stand in the same column. Subtract as 
68.625 in integers, placing the decimal point on the left of the 

; tenths in the remainder. 


168.875 NorTre.—In the process, the .5 have been reduced to .500 by an- 
“O00 nexing two 0’s. Itis not necessary, however, to write the 0’s. 


Rule.— Write the numbers so that figures of the same 
order shall stand in the same column. Add or subtract as 
in integers. Place the decimal point at the left of the tenths 
in the result. 


9. Add 83.4, 9.64, 324.5, and .8946. 
10. Add .375, .0049, 6.2, 54., and 3.03. 
11. Subtract 23.45 from 198.62. 
12. Subtract 5.946 from 594.6. 
13. From the sum of 845, 9.64, and .04, subtract the 
sum of 4.5 and 50.656. 


160. In United States money, the dollar is the wnit ; 


148 ELEMENTARY ARITHMETIC. 


cents are hundredths, and occupy the two places on the 
right of the decimal point; mills are thousandths, and oc- 
cupy the third decimal place. The sign of U. 8. money 
is $. ; 
Thus—One dollar is written - - - $1.00 or $1. 


One cent is written - ~ - - OL oracle 
Ten cents are written - - - .10 or 10¢ 
One mill is written = - - - - .OOL 
Ten dollars, thirty-seven cents and five 

mills are written — - - - - $10.875 or $10 387}. 


(For fractional parts of a dollar, see page 95.) 


FRACTIONAL PARTS OF A CENT. 


oO 
oO 
5 
ct 
| 


+ cent =$.005. 4 $.003$. 
4 cent = .0025. 4 cent = .0018. 


LZ. Add $25.50, $8.75, $492.00, and $3.25. 

15. Add $50., $125.875, $.625, and $35.00. 

16. Add $310., $3.10, $49., $.10, and $.007 

17. Add $74, $34, $104, and $3. 

SuGGEsTION.—Change the fractions to decimals, in accordance 
with the table, page 95. Thus, $74=$7.50; $8£= $3.25; etc. 

1S. Add $247.50, $29.75, $84, and $932. 

19. Find difference of $100. and $753. 

20. Add $1.374, $2.25, $.125 and $35. 

SucGEsTron.—Change the fractions of a cent to mills. 

21. Add $.50, $.80, $.064, $.374, and $.1675. 

22, From $254 subtract $15.62. 

23, A horse was bought for $150. and sold for $170.50. 
What was the gain? 

24. A man bought a load of hay for $18.75, another for 
$13.25. He sold the two loads for $35. Did he gain or 
lose, and how much? 

25. A house was bought for $3420.75. For how much 
must it be sold to gain $1753? 


DECIMAL FRACTIONS. 149 


26. John spent 85 cents for a knife, one dollar and a 
half for a sled, and a quarter of a dollar for a ball. What 
change should he have received from a $5 bill? 


MULTIPLICATION OF DECIMAL FRACTIONS. 


ORAL. 


161. 1. 5 times .1 are how many tenths? 

2, 3 times .3 are how many tenths? 

3. 4 times .6 are how many tenths? 

SoLuTrion.—4 times .6 are 24 tenths, equal to 2 units and 4 tenths, 
or 2.4, 

4. Multiply .8 by 6; by 7; by 8. 

5. Multiply .09 by 5; by 6; by 7. 


Note.—In decimals, as in integers, the product is the same denomination, or 
kind, as the multiplicand. 


6. Multiply 1.2 by 3; by 4; by 5. 

7. Multiply .1 by .1. 

Sotution.— .1X 1= 545 X 745 = zis that is, .01. 
S. Multiply .3 by .03. 


SOLUTION.— .3X.08=-3, X zag aya, OF -009. 


Notre I.—In the multiplication of one decimal fraction by another decimal 
fraction, there is really the same process as in the multiplication of any fraction by 
another fraction; that is, the numerator of the product is the product of the 
numerators of the factors; and the denominator of the product is the product of 
the denominators of the factors. Since the denominator of every decimal fraction 
is a power of 10, the product of the denominators is that power of 10 which con- 
tains as many 0’s as both, or all, of the denominators of the factors. Thus, 


io XPo=Tb0 To X rho too} Too X Tod THbo: Me. In mult 
plying one decimal fraction by another, therefore, itis necessary to find the product 
of the nwmerators only, and make the number of decimal orders in the product of 
the numerators equal to the number in both of the factors. Thus, .3x.5=.15. 

Nore IT. — It is frequently necessary to prefix 0’s tothe product of the numera- 
tors, in order to secure the requisite number of decimal orders. Thus, .1 x.1=.01; 


-01 x .01=.0001; ete. 


PRINCIPLE. 
3. The number of decimal orders in the product of two 
or more decimal factors is always equal to the number 
of decimal orders in all the factors. 


150 HLEMENTARY ARITHMETIC. 


WRITTEN. 
9. Multiply .325 by 5. 
PROCESS. 
“329 SoLution.— 5 times 3825=5 times 332°, =—1$23= 
1.695. 
1.625 
‘10. Multiply .325 by .05. 
PROCESS. 
320 SoLuTiIon.— .825 x .05 = fxs x spy — res —= 
O05 91625. 
.01625 


Rule.—Multiply as in integers, and make the number 
of decimal orders in the product equal to the number of 
decimal orders in both factors. If necessary, prefix 0's. 


Multiply: 
11, Av by 8. 16. .0324 by .25. 
12, 1.35 by 24. 17. 9873 by .041. 
13, 38.96 by 1.56. 18. 50.94 by 334. 
14, .1824 by .15. 19. $3827.50 by 24. 
15. 2.549 by .0382. 20. $84.675 by 12. 


21. Find cost of 25 yards of cloth, worth $4.25 per yard. 

22, Find value of 3625 bushels of wheat, worth $1.124 
per bushel. 

23. aa cost of 2450 acres of land, worth $28.50 per 
acre. 

24. Multiply 1.8456 by 100. 

SuGGESTION.—Simply remove the decimal point two places to the 
right. 

25. Multiply 93.426 by 1000. 

26. Find product of .18433 by 100. 

27. Multiply .4 by 100.: 


Suee@Estion.—Removing the decimal point one place to the right 


DECIMAL FRACTIONS. 151 


multiplies by 10; erasing this point and annexing 0 multiplies again 
by 10. 

28. Multiply 3.5 by 100. 

29. Multiply 54.69 by 1000. 


DIVISION OF DECIMAL FRACTIONS. 


ORAL. 


162. 1. Divide 6 by 2. 
SoLutTion.—Dividing by 2 is finding one half. One half of .6 is .3. 


2. Divide .8 by 2; by 4. 

3. Divide .12 by 2; by 3; by 4. 

4. Divide 3.6 by 9. 

SoLtutron.— 3.6=36 tenths. +4 of 36 tenths is 4 tenths, or .4. 


5. Divide 5.6 by 8; by 7. 
6. Divide 2.4 by 3; by 4; by 6; by 8. 
7. Divide 4.8 by 4; by 6; by 8; by 12. 


163. The difficulty in division of decimals is the placing 
of the decimal point in the quotient. But it should be borne 
in mind that division is the converse of multiplication; that 
the divisor and quotient are factors of the dividend. Hence, 
the number of decimal orders in the divisor and quotient 
must be equal to the number of decimal orders in the divi- 
dend. ‘Therefore, the decimal orders in the quotient must 
equal the excess of the decimal orders in the dividend over 
those in the divisor. 

That is, if there are 5 decimal places in the dividend, and 2 dect- 
mal places in the divisor, there must be 3 decimal places in the 
quotient. 

In each of the three preceding examples, there is one decimal place 
in the dividend, none in the divisor; therefore, there must be one 
decimal place in the quotient 


NoTe.—The correctness of the pointing in the quotient may always be tested 
by proving the work; since the dividend must be the product of divisor and quotient. 


152 HLEMENTARY ARITHMETIC. 


PRINCIPLE. 


4. The decimal orders in the divisor and quotient are 
equal in number to the decimal orders in the dividend. 


164. The division of a decimal fraction or mixed num- 
ber by a decimal fraction may be considered in three cases. 

1. When the decimal orders in the dividend and divisor 
are equal in number. 

2. When the decimal orders in the dividend are Jess in 
number than those in the divisor. 

3. When the decimal orders in the dividend exceed in 
number those in the divisor. 


CASE I. 


165. 1. Divide .9 by .3. 


SoLution.— 9-+-.3=795+7%5=9+3=3. 


2. Divide .7 by .3. 
3 


SoLuTion.— .7+.8=75+335=7+3=24. The common fraction + 
may be changed to a decimal (Art. 157). 


NotTE.—When dividend and divisor are fractions having a common denom- 
inator, their quotient is the quotient of the numerator of the dividend divided by 
the numerator of the divisor (Art. 148). When dividend and divisor are decimal 
fractions having the same number of decimal orders, they have a common denom- 
inator; and their qrvotient is the quotient of their nwmerators, and is an integer or a 
mixed number. When the dividend is a multiple of the divisor, the quotient is an 
integer. Thus, 2.4+.8=38. When the dividend is nota multiple of the divisor, the 
quotient isa mixednumber. Thus, 2.5+.8=34, or 3.125. 


Divide .8 by .4. 7. Divide 3.6, by.-9. 
Divide .6 by .2. S. Divide 7.2 by .8. 
Divide .18 by .06. 9, Divide 1.44 by .12. 
Divide .24 by .08. 10. Divide 1.21 by .11. 


TD OW Ss 


Lf. Divide .21 by .04. 
SoLutTion.— .21-++-.04=9%))5 +745 = 21+4=5¢ or 5.25. 


12. Divide .45 by .06; by .07; by .08; by .09. 


DECIMAL FRACTIONS. 153 


CASE II. 


166. 1. Divide .8 by .04. 

So_uTrion.—Annex a sufficient number of 0’s to the dividend to 
make the decimal places in the dividend equal to those in the divisor. 
Then proceed as in Case I. .8=.80. .80+.04= 89, + 75 =80+4=20. 

2. J)ivide .9 by .004. : 

SoLuTIon.—.9=.900; .900-+.004=395°9'5 + app 7 = 900-+4 = 225. 

Divide: 


Soo by +03. 6, .2 by .04. 9. 3d by .002. 

oO by 02. fea Dye US 10. «7 by .004, 

Goel by Uo. &. .3 by .06. if, 12 by 006: 
CASE III. 


167. 1. Divide .24 by .3. 


SOLUTION.— 24+.3 =?~5+ fin = tit Ms 1.0 = ==, 

Norse.—The denominator of the quotient is the quotient of 100+10=10; that is, 
the denominator of the quotient contains as many 0’s as the 0’s in the denominator 
of the dividend exceed the 0’s in the denominator of the divisor. Or, when the 
fractions are written decimally, the quotient contains as many decimal places as the 
decimal places in the dividend exceed those in the divisor. 


Divide: 

2. .35 by .7 6. .144 by .12. LO, 0044 by .11. 
3. A2by 6. 7. 80 by 8. 11. .096 by .12. 
4, .81 by .9 8. .056 by .7. 12, .0024 by .003. 
d. .63 by .? 9. 0072 by .08. 13. .0049 by .007. 


DIVISION OF DECIMAL MIXED NUMBERS. 


168. 1. Divide 5.12 by .8. 


ANALYsIs.—Regard both numbers as integers; then 512+8=64. 

Dividing the divisor, 8, by 10, multiplies the quotient by 10; that is, 
512+.8=640. 

Dividing the dividend, 512, by 100, divides the quotient by 100; that 

5.12+.8=6.40, or 6.4. 

Observe that the number of decimal orders in the divisor and 
quotient equals the number of decimal orders in the dividend. 


154 ELEMENTARY ARITHMETIC. 


2. Divide 28.4967 by 5.84. 


PROCESSES. 
5.84)28.4967 (4.87 + 
2336 
5136 
4672 
SoLtutTion.—Since there are two decimal 
4647 orders in the divisor, and four in the divi. 
4088 dend, there will be two decimal orders in 
the quotient. The remainder is 559. If the 
559 division is continued by annexing 0’s to 
the dividend, there will be one decimal 
as ai figure in the quotient for every 0 annexed 
- to the dividend. 
6136 NOTE I.—Annexing 0 toa remainder is equiva- 
46 12 lentto annexing it to the dividend, and then “ bring- 
ing it down.” 
4647 NOTE II.—Incomplete division is usually indi- 
4088 cated by annexing to the quotient the sign +. 
9090 
5206 
3040 
2920 
420 
3. Divide .48756 by 324.5. 
PROCEsS. 
324.5).48756(.0015 + SoLuTron.—Since there are five decimal 
3945 orders in the dividend, and one in the 
eee divisor, there must be four in the quotient. 
16306 But since the quotient contains but two 
16225 figures, 15, it is necessary to prefix (not 
ee ee annex) two 0’s. 
81 


Rule.—Divide as in integers. Make as many decimal 
orders in the quotient as the number of decimal orders in the 


DECIMAL FRACTIONS. 155 


dividend exceeds the number of decimal orders in the divisor. 
If necessary, prefix 0's to the significant figures of the 
quotient. 


Suaarsitions.—/f the decimal orders tn divisor and dividend are 
equal in number, the quotient is an integer, mixed number or fraction. 

If the decimal orders in the dividend are less in number than those 
in the divisor, make them equal by annexing 0’s to the dividend. 

Before dividing, ascertain how many decimal places there should be 
in the quotient. When the last figure of the dividend has been used, 
place the decimal point in the quotient, prefixing 0's if necessary. Con- 
tinue the division, tf desirable, by annexing 0’s to the aividend. 


Divide: : 

4. 846.75 by 4.5. 11. 5434.96 by 82.451. 
}. V3A484 by 3.28. 12, 723.45 by .3464. 
6. 10.0946 by .23. 13, .95643 by 78.4. 
7. 64.756 by .834. 14. 75678. by 0375. 
S. 7.5683 by 1.34. 15. 95.698 by .0675. 
9. 946.75 by 3.475. 16. 3.6482 by .00045. 

LO. 83.7456 by .845. 17. 45.836 by 2.3049. 


18. Divide 438.467 by 100. 


Soturron.—Remove the decimal point two places to the left. 
438.467 + 100=4.38467. 


Divide: , 
19. 348.75 by 100; by 1000; by 10000. 
20. 5096.1 by 100; by 1000; by 10000. 
21. 8 by 10; by 100; by 1000. 


22. If a locomotive run 62.5 miles in 2.5 hours, what is 
its speed per hour? 

25, What is the cost of one acre of land, if 19.5 acres 
cost $73.125? 

24. What number, multiplied by 28.75, will produce 
672.75? 

25. If a ship sail 431.25 miles in 37.5 nours, what is her 
average speed per hour? 


156 ELEMENTARY ARITHMETIC. 


26. Find value of one ox, if 3874 oxen are worth 
$11921.25. 

27. One degree of latitude is about 69.16 miles. How 
many degrees of latitude are there in 24897.6 miles? 


REVIEW QUESTIONS. 


How are decimals written for addition and subtraction? "What is 
the unit in United States money? What part of a dollar is a cent? 
What part of a dollar is a mill? How many decimal orders in the 
product? How many decimal orders in the quotient? How may the 
correctness of the pointing in the quotient be proved? When the 
decimal orders in the divisor and dividend are equal in number, what 
is the quotient? What should be done when the decimal orders of 
the dividend are less than those in the divisor? What, when the 
decimal orders in the dividend exceed those in the divisor ? 


MISCELLANHOUS PROBLEMS. 


1. Add fifteen thousandths, thirty-one hundredths, 
one thousand and twenty-nine millionths, eighty-one ten- 
thousandths, three hundred and twenty-seven, seven tenths. 

2, Add $84.75, $3.874, $40.09, $.75, $.08, $5.33, $196.45, 
$33.62. 

3. From 9 times 345.67 subtract 984.64 divided by 8. 

4, (434.55--5) + (4.847 x .009) — (2.52 + .03) =? 

5. Reduce to decimals and find sum of $, #, 2, 4, s'y, a- 

6. Reduce to common fractions in lowest terms: .6, 
379, .0625. 

7. Divide 846.7896 by .06. 

8. Divide 925. by .00025. 

9. Divide .375 by 125. 

10. Multiply 98.34 by .0078. 

11. A man earned $16.25 one week, $25.625 the next 
week, $18.00 the third week, and $21.375 the fourth week. 
His expenses averaged $12.25 per week. How much money 
did he save in the 4 weeks? 


DECIMAL FRACTIONS. 157 


12. An agent traveled 324.8 miles in January, 496.6 in 
February, 134.84 in March, 740.38 in April. How many 
miles did he average per month? 

13. Mr. C. had a farm of 1800 acres. He sold 134.75 
acres to Mr. D., 86.5 to Mr. E., 240.08 acres to Mr. F., and 
half of the remainder to Mr. G. How much remained? 

14. If a ship averages 135.663 miles per day, how many 
miles will it sail in 2 weeks and 4 days? 

15. Find cost of shingles for 6 houses, if each house 
needs 7500 shingles, worth $4.25 per M. 

16. Find cost of 22480 bricks at $9.25 per M. 

17. How many feet of boards, at $12.50 per M., can be 
bought for $862.50? 

18. How many bricks, worth $8.625 per M., can be 
bought for $603.75? 

19, How many bushels of oats, worth .375 per bushel, 
are equal in value to 62.5 bushels wheat, worth $1.125 per 
bushel? 

20. Divide 84.35 by 52.7185; multiply the quotient by 
003. Find sum of the two results. 

21. Mr. C.’s expenses were $80.75 in January, $75.375 
in February, $64.625 in March, $92 in April. What did his 
expenses average per month? What were his expenses for 
a year, at the same rate? 

22, What is the value of a farm if .15 of it be worth 
$2250? 

23, 270 is 2 of what number? 

24. 270+1.124=? 

25. 270 is 1.124 of what number? 

26. 500 is 1.25 of what number? 

27. A merchant sold some dry goods for $525. which 
was 1.05 of the cost. What was the cost? 

28. Mr. D. sold an engine for $9405, which was 1.374 of 
its cost. What was the cost of the engine? 

29. 620 is $ of what number? 

30. 620 is § of what number? 


158 ELEMENTARY ARITHMETIC. 


G1, 1440+.75=? 

32. Mr. E. sold a house for $2880, which was .75 of its 
cost. What did the house cost? 

33. Mr. F. sold a farm for $7850, which was .874 of its 
cost. What was the cost? 

34. Find 82 hundredths of 284.75. 

36. Find 124 hundredths of 54678. 

36, (48.75 x .15)+4(65.78 x .75) =? 

37. Mr. G. bought oxen for $3150; he sold them for 
$2800, and thereby lost $5 on each ox. How many oxen 
did he buy? 

38. Mr. H. sold 324 sheep for $1134, and thereby gained 
$1.50 on each. What did he give for them? 

39. Mr. J.’s house cost $4500, which was 3 times the 
cost of his furniture, and twice the cost of his lot. Find 
cost of all. 

40, At $5.75 per cord, how many cords of wood may be 
exchanged for 2875 bushels of oats worth 373 cents per 
bushel? 

41. A man gave 324.75 bushels of potatoes in exchange 
for 30 calves, valued at $129.90. What did each calf cost 
and what was the value of the potatoes per bushel ? 

42, 125X2+2.5=? 

43. $X.375+2=? 

44, A merchant sold 5 bales of cotton at $17.75 per 
hundred pounds; one bale contained 452.85 lb., the others 
324.92 lb. each. What did he receive for them all? 

45. Find four equal numbers the sum of which is 324.125. 

46. The sum of two numbers is 535.75; one of them is 
228.496. What is the other? 

47. The difference of two numbers is 8.25; the less 
number is four times the difference. What is the sum of 
the two numbers? 

48. The product of three numbers is 39.33; one of the 
numbers is 13.8; another is 7.125. What is the third 


number? 


>) 


DENOMINATE NUMBERS. 159 


ee ERO 21" 


DENOMINATE NUMBERS. 


Art. 169. Numbers are either Abstract or De= 
nominate. 

170, An Abstract Number is a number whose 
kind of unit is not named. 

171. A Denominate Number is a number whose 
kind of unit is named. 


Thus, 1, 3, 10, are abstract numbers; 1 dollar, 3 mills, 10 horses, 
are denominate numbers. 

172. ASimple Denominate Number expresses 
units of but ove denomination.. 

Thus, 5 cents, 2 quarts. 

173. A Compound Denominate Number 
expresses units of two or more denominations of the same 
scale. 

Thus, 2 feet 8 inches; 8 pounds 10 ounces. 

174. <A Solid is a body which has three dimensions, 
length, breadth and thickness. 

A book is a solid; so is a marble. 

175. A Surface has two dimensions, length and 
breadth. 


This /eaf is a solid, because it has length, breadth and thickness. 
This page (one side of the leaf) is a surface, because it has length and 
breadth, but not thickness. 


160 ELEMENTARY ARITHMETIC. 


176. A Line has one dimension only, length. 


A line is the measure of distance. This page is 8 inches long. The 
lines drawn on the black-board have both breadth and thickness; but 
they are not mathematical lines. Even the lines drawn by the finest 
lead pencil have breadth and thickness. They must have breadth, in 
order to be seen: but the essential part of a line is its length. 


MEASURES OF LENGTH. 


177, Linear measures are used in measuring distances. 
The dimensions of surfaces and solids are found by measur- 
ing the distances between certain points. The length of a 
brick is the distance from end to end. 


TABLE OF LINEAR (OR LONG) MEASURE. 


12 inches (in.) = _ 1 foot (ft.) 
3 feet = 1 yard (yd.) 
ee = 1 rod (rd.) 

163 feet 

320 rods _= 1 mile (m.) 
ORAL. 


1, How many inches in 3 feet? 

ANALYSIS.—In 8 feet there are 3 times as many inches as in 1 foot. 
In 1 foot there are 12 inches; in 3 feet there are 3 times 12 inches, or 
36 inches. 

2, How many inches in 2 feet? In 4 feet? In 7 feet? 
In 8 feet? 

3. How many feet in 3 yards? In 5 yards? In 6 
yards? 
4, How many rods in 2 miles? In 3 miles? 
5. How many feet in 36 inches? 


ANALYsIs.—Since there are 12 inches in 1 foot, in 36 inches there 
are as many feet as 12 are contained times in 36, or 3 feet. 


6. How many feet in 24 inches? In 72 inches? In 96 
inches? In 30 inches? In 50 inches? 


DENOMINATEH NUMBERS. 161 


7. How many yards in 6 feet? In 18 feet? In 20 
feet? In 25 feet? 
5. How many inches in 2 yards? 
SuaaEstion.—Find the number of inches in 1 foot; in 1 yard; in 
2 yards. 
9. What is the cost of 4 yards of ribbon, worth 6 cents 
per yard? 
ANALYsIS.— 4 yards cost 4 times as much as 1 yard. 1 yard costs 
6 cents; 4 yards will cost 4 times 6 cents, or 24 cents. 


10. Find cost of 8 yards of muslin, at 10 cents per yd. 

11, What is the cost of 7 yards of cambric, at 12 cents 
per yd.? 

12. How many inches in of a foot? 

Anatysis.—In one half a foot there are one half as many inches 


as in one foot. In one foot there are 12 inches; in 4 of a foot there 
are 4 of 12 inches, or 6 inches. 


13. How many inches in } of a foot? In 4 of a foot? 

14. How many rods in $ of a mile? In} of a mile? 

15. How many inches in 3 of a foot? In 3 of a foot? 

SueGrEstrons.—There are 3 times as many inches in # of a foot as 
there are in ¢ of a foot. 

16. How many inches in 1} feet? In 32 feet? 

17. How many feet in 24 yards? In 5% yards? 

1S. What part of a foot is 1 inch? 6 inches? 4 inches? 

19, What part of a yard is 1 foot? 2 feet? 

20. What part of a yard is 1 inch? 6 inches? 9 inches? 


WRITTEN. 


21. How many inches in 400 yards? 


Suaarstrons.—Reduce to feet; then to inches; or, reduce to inches 
at one operation, thus: Since there are 36 inches in L yard, in 400 
yards there are 400 times 36 inches; 3640014400 inches. 


22. How many inches in 328 yards? In 540 yards? 


162 ELEMENTARY ARITHMETIC. 


23. Wow many feet in 84 rods? In 250 rods? 

24. How many yards in 3 miles? In 20 miles? 

25. How many inches in 50 rods? In 160 rods? 

26. What part of a mile is 1 yard? 

27. What part of a rod is 1 inch? 

28. What part of a mile are 20 inches? 5 feet? 

29. Find cost of 256 yards cloth at $3.50 per yd. 

30. Find cost of 596 feet of rope, at 24 cents per ft. 

31. How many miles can a horse travel in 12 hours, if 
he travels 3% miles in one hour? 

32. A locomotive ran 512 miles in 16 hours. What was 
its speed per hour? 


MEASURES OF SURFACE. 


178. A Plane Surface is a surface which does 
not change its direction. 

A slate is a plane surface; floors and ceilings are plane surfaces. 
The surface of a ball is not a plane surface, because the surface con- 
stantly changes its direction. 


179. A Plane Figure is a portion of a plane sur- 
face. 

A pane of glass in a window is a plane figure. If the sides of the 
pane are four straight lines, it is a guadrilateral. The corners of the 
pane of glass are called angles. If the pane has four corners, or 
angles, and they are equal to each other, they are right angles, and the 
pane is a rectangle. 

A slate is usually a rectangle; its corners, or angles, are right 
angles. 

180. A Quadrilateral is a plane figure bounded 
by four straight lines. 

181, An Angle is the divergence of two straight 
lines which meet or intersect. 


The angle A C B is formed by the 


A 
eee ree: meeting of the lines A C and B C at 


Gor 5 ee the point C. 


DENOMINATE NUMBERS. 


163 
182. 


A Right Angle is one of the four equal 


angles formed by the intersection of two straight lines. 


A right angle is often one of the two equal angles formed 
by one straight line meeting another straight line. 


The four angles at the point 0 are right 
angles. 


183. A Rectangle is a quadrilateral whose angles 
are right angles. 


When the sides of a rectangle are equal, the rectangle is a 
square. 


184. A Square is a rectangle having equal sides. 


A square inch is a square, one inch long 
and one inch wide. 


a 
2 ; 
| 1 square inch. 
rr 
* 


eines 
185. Surface, or area, is measured by square units. 


A piece of paper which may be cut, without waste, into three 
pieces, each one square inch in size, contains three square inches. 


The area of floors and walls may be expressed in square feet or in 
square yards. 


Farms are usually described as containing a certain number of 
acres; as 80 acres, 160 acres, ete. 
Counties, states, and other large portions of the earth’s surface, are 
measured by square mes. 


186. Area is considered a product, of which length 
and breadth are the factors. 


The number of square inches 
ina square foot is the product of the length and breadth of 


the square foot, expressed in inches: 12x 12=144. 


164 ELEMENTARY ARITHMETIC. 


187 TABLE OF SQUARE MEASURE. 


square foot (sq. ft.) 
square yard (sq. yd.) 
square rod (sq. rd.) 
acre (A.) 


square mile (sq. m.) 


144 square inches (sq. a) = 

9 square feet . S 
304 square yards - = 
160 square rods - 
640 acres - - 


fet pt pl ee 


ORAL. 


1. How many square inches in 2 sq. ft.? 

2, How many square yards in 54 sq. ft.? 

3. How many square feet in 4 ae rds. ? 

4. How many square inches in $ of asq. ft. In} of a 
iq. {t.? In # of a sq. ft.? 

5. How many sq. rods in 4 of an acre? 

6. How many acres in $ of a sq. mile? 

7. Why are there 304 square yards in a square rod? 

SuGGESTION.—See table of linear measure. 


S. How many sq. ft. in 3 of a sq. yd.? 


Notes I.—This diagram illustrates the 
fact that the square of 12 is 144; the square 
of 12 linear inches is 144 square inches. 

Nore II.—The line A E is one half of 
the line A B; but the square A E F G is one 
fourth of the square ABCD. That is, the 
square of one half of a line is one fowrth of 
the square of the whole line. So, the square 


of 4 is 4; : etc. See Art. 143. 


9. How many inches in 4 of a sq. ft.? 
10. How many inches in the square of $ of a foot? 
11. A man having a field containing 144 sq. rods sold 36 
sq. rods. What part of his field did he sell? 
12. What part of a sq. yd. is a sq. ft.? 


TS. 
nicnes? 
14. 
Ree 
16. 
rie 


18. 
19. 
20. 
21, 
23. 
24, 
25, 
26. 
QT. 
28. 
29, 
30. 
$1, 


DENOMINATE NUMBERS. 165 


What part of a square ft. are 36 sq. inches? 72 sq. 
108 sq. inches? 

What part of a sq. foot is the square of 4 of a foot? 

What part of a sq. foot is the square of 3 inches? 

What part of a sq. foot is the square of 4 of a foot? 

What part of a square mile are 160 sq. rods? 


WRITTEN. 


How many sq. inches in 29 sq. ft.? 

How many sq. ft. in 9324 sq. yds.? 

How many sq. ft. 43200 sq. inches? 

How many sq. yards in 3042 sq. feet? 

How many sq. miles in: 3200 acres? 

How many acres in 329 sq. miles? 

How many sq. yards in 5 acres? 

How many sq. rods in 16 miles? 

How many sq. rods in 588060 sq. ft.? 

How many acres in 2178000 sq. ft.? 

How many sq. inches in | acre? 

What part of 1 sq. mile is 1 sq. rod? 

What part of 1 sq. mile are 25 sq. rods? 

What part of 1 sq. yd. is 1 sq. inch? 

What part of 1 sq. yd. is the sq. of $ of a sq. ft.? 
Draw a diagram illustrating the difference between 9 


syuare feet and 9 feet square. 


ANGULAR OR CIRCULAR MEASURE. 


#88. <A Cirele is a plane figure bounded by a line 
every point of which is equally distant from a point within 
the circle, called the center. 

189. A Circumference is the line which bounds a 


circle. 


190. An Are is any part of a circumference. 


166 ELEMENTARY ARITHMETIC. 


191. A Degree is one of the 360 equal parts of a 
circumference. 

Every circumference contains 360 degrees; the size of a degree, 
therefore, depends upon the size of the circumference of which it is 
a part. 

192. A Diameter is a straight line extending 
through the center of a circle, terminating in opposite 
points of its circumference. 

193. A Radius isa straight line extending from the 
center of a circle to any point in its circumference. A 
radius is one half of a diameter of the same circle. 

194. An arc measures the angle formed by two radii 
drawn to its extremities. 


The arc D E measures 
B the angle DCE. Each is 
45 degrees. 


DIAMETER 


Cc e 
‘ROUMFEREN? 


198. TABLE OF ANGULAR OR CIRCULAR MEASURE. 


60 seconds (") = 1 minute (’) 

60 minutes = 1 degree (°) 

90 degrees =} 1 right angle. 
1 quadrant. 


360 degrees = 1 circumference (cirf.) 


NoTE.—A degree on the equator is 694 miles. 


DENOMINATE NUMBERS. 167 


ORAL. 


1. How many seconds in 2 minutes? In 3 minutes? 


In 5 minutes? 
2. How many minutes in 4 degrees? In 6 degrees? 


In 8 degrees? In 10 degrees? 


3. How many seconds in 4 of a minute? In 4 of a 
minute? In 54; of a minute? 
4. How many minutes in 3 of a degree? In 3 of a 


degree? In 4 of a degree? 
5. How many minutes in 24 degrees? 3% degrees? 
52 degrees. 
6. How many minutes in 120’? In 360°? In 480"? 
7. How many degrees in 240’? In 180°? In 720’? 
S. How many degrees in 3600’? 
9. How many degrees in 1 right angle? In 2 right 
angles? 
10. How many quadrants are equal to 3 right angles? 
J1. How many quadrants in a circumference? 
12. How many right angles at the center of a circle? 
13. How many degrees in 34 right angles? 
14. How many degrees in one sixth of a circumference? 


NOTE.—One sixth of a circumference is called a Sextant. 


15. How many degrees in 2 sextants? 

16. How many quadrants are equal to 3 sextants? 

17. How many right angles in 44 sextants? 

18. How many degrees in one twelfth of a circum- 
ference? 


NorTe.—One twelfth of a circumference is called a Sign. 


19, How many degrees in 2 signs? In 3 signs? In 5 
signs? 

20. How many signs in 1 circumference? In 1 sextant? 
In 1 quadrant? 

21. How many signs in 180°? How many sextants? 
How many quadrants? How many right angles? 


168 ELEMENTARY ARITHMETIC. 


WRITTEN. 


22, How many degrees in 320’? 

23. How many minutes in 480"? 

24, Wow many degrees in 7200"? 

25. How many seconds in 53 degrees? 

26. How many seconds in one circumference? 

27. How many degrees in 28800"? 

28. A degree of the equator is 694 miles. What is the 
length of the equator in miles? 

29, What is a quadrant of the earth’s equatorial circum- 
ference? | 

30. A ship has sailed 129 degrees. How many miles 
has she sailed, if each degree equals 543 miles? 

ol. ‘Thirty degrees of a hoop measured 60 inches. What 
was the length of the hoop in feet? 

o2, If a quadrant, a sextant, and a sign, be taken from 
a circle, how many degrees will remain? 


MEASURES OF VOLUME. 


196. A Solid isa body which has three iar 
length, breadth, and thickness. 


Bricks, boards, balls, stones, etc., are solids. Water is called a solid, 
because it has three dimensions; the swrface of water is a plane, 
having but two dimensions. 


197. A Cube is a solid having six equal square sides 
or surfaces. 


a 
ea 
ye 


A cubic inch is a solid 1 inch long, 1 inch 
wide, 1 inch thick. It has six surfaces, each 
1 inch square. It has 12 edges, each 1 inch 
long. 


198. Solidity, or volume, is measured by units of volume, 


DENOMINATE NUMBERS. 169 


or cubical units. A room is said to contain a certain 
number of cubic feet; a gallon contains 231 cubic 
inches; etc. 

Volume is a product, of which length, breadth, and thick- 
ness, are the factors. The number of cubic inches in a 
cubic foot is the product of the length, breadth, and 
thickness of a cubic foot expressed in inches: 1212 x 
12=1728. 


199. TABLE OF CUBIC MEASURE. 
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.) 
27 cubic feet = 1 cubic yard (cu. yd.) 
247 cubic feet = 1 perch of stone or masonry. 
128 cubic feet = iecordi{cd:) 
ORAL. 
7. How many cubic feet in 2 cubic yards? 
2. How many cubic inches in $ of a cu. ft.? 
3. How many cubic feet in $ of a perch of stone? 
4. How many cubic feet in 4 of cord? 


WRITTEN. 


5. How many cu. in. in 36 cu. ft.? 
6. How many cu. feet in 691200 cu. in.? | 
7. How many cu. feet in 428 cords? 
S. How many cords in 64000 cu. ft.? 
9. How many cu. ft. in 40 perch of stone? 
10. How many perch of masonry in 24750 cu. ft.? 
11. How many cu. inches in 1 cord? 
12. What part of a cu. foot is a cu. inch? 
13, What part of a cord is 1 cu. yard? 
14. What part of 5 cords are 12 cu. ft.? 


200. Liquid measures are used in measuring liquids. 


170 ELEMENTARY ARITHMETIC. 


TABLE OF LIQUID MEASURE. 


4 gills (gi.) = 1 pint (pt.) 

2 pints = 1 quart (qt.) 

4 quarts = 1 gallon (gal.) 
31} gallons = 1 barrel (bbl.) 


63 gallons (2 bbls.) 


Norre.— 1 gallon=231 cubic inches. 


1 hogshead (hhd.) 


ORAL. 


1. How many gills in 7 pints? In 10 pints? In 20 
pints? 
2. How many pints in three quarts? In 5 quarts? In 
8 quarts? 
3. How many quarts in 5 gallons? 9 gallons? 12 gal- 
lons? 
4. How many pints in 40 gills? In 80 gills? In 100 
gills? 
5. How many quarts in 24 pints? In 36 pints? 
6. How many gallons in 16 quarts? In 44 quarts? In 
60 quarts? 
7. How many gallons in 2 barrels? In 10 barrels? 
S. How many quarts in one barrel? 
9. How many barrels in 7 hogsheads? In 9 hogsheads? 
10. How many hogsheads in 12 barrels? In 30 barrels? 


11. How many cubic inches in 1 gallon? In 2 gallons? 


WRITTEN. 


12. How many pints in 3460 gills? 
13. How many quarts in 1824 pints? 
14. How many gallons in 8 barrels? 
15. How many quarts in 2 barrels? 
16. How many pints in 327 quarts? 
17. How many pints in 10 hogsheads? 


SucGEstTions.—First find the number of gallons in 10 oes Rats 
then change the gallons to quarts, etc. 


18. 
19. 
20. 


21 


DENOMINATE NUMBERS. 


How many gills in 20 barrels? 

How many gills in 3 hogsheads? 
How many cubic inches in 8 gallons? 
How many cubic inches in 1 barrel? 


171 


201. Dry measures are used in measuring grain, 
fruit, ete. 


TABLE OF DRY MEASURE. 


2 pints (pt.) = 1 quart (qt.) 
8 quarts = 1 peck (pk.) 
4 pecks = 1 bushel (bu.) 


Nore.— 1 bushel=2150.4 cubic inches. 


S © WD AP So ON 


ORAL. 


How many pints in 16 quarts? In 25 quarts? 
How many quarts in 40 pints? In 48 pints? 
How many quarts in 3 pecks? In 8 pecks? 
How many pecks in 32 quarts? In 40 quarts? 
How many pecks in 5 bushels? In 7 bushels? 
How many bushels in 24 pecks? In 36 pecks? 
What part of a peck is a quart? 

What part of a bushel is a peck? 

How many quarts in 1 bushel? 

What part of a bushel is 1. quart? 3 quarts? 


WRITTEN. 


How many quarts in 329 pecks? 

How many bushels in 444 pecks? 

How many quarts in 72 bushels? 

How many bushels in 12800 quarts? 

How many pecks in 6400 pints? 

What part of a bushel is 1 quart? 

What part of a bushel are 3 pints? 

How many cubic inches in 3 bushels? 
How many cubic inches in 1 peck? 

How many bushels in 21504 cubic inches? 


172 ELEMENTARY ARITHMETIC. 


MEASURES OF WEIGHT. 


202, Avoirdupois weight is used in weighing merchan- 
dise that is bought and sold by weight, such as groceries, 
coal, ete. 


TABLE OF AVOIRDUPOIS WEIGHT. 


16 ounces (0z.) = 1 pound (Ib.) 
100 pounds = 1 hundredweight (cwt.) 
20 hundredweight = ta ton (T.) 
2000 pounds = 


NoTe.—The “long ton,” containing 2240 1b., is used in Great Britain, in United 
States Custom Houses, and in Pennsylvania coal mines. 


ORAL. 


How many ounces in 3 pounds? In 5 pounds? 
How many pounds in 32 ounces? In 64 ounces? 
How many ounces in 1 cwt.? In 2 ecwt.? 

How many pounds in half a ton? 

How many pounds in a quarter of a cwt.? 

How many cwt. in one tenth of a ton? 

What part of a pound are 5 ounces? 


BO etd een SSO! 


What part of a cwt. is one ounce? 


WRITTEN. 


9. How many lb. in 17 tons? 
10. How many ewt. in 25600 ounces? 
11. How many tons in 598000 lb.? 
12. How many ounces in 3240 lb.? 
18. How many lb. in 38 cwt.? 
14. How many oz. in 2 T.? 
15. What part of 1 cwt. is 1 o2z.? 
16. What part of 1 T. is ‘1 lb.? 


203. Troy weight is used for Weleane gold, silver, 


and precious stones. 


DENOMINATE NUMBERS. 178 


TABLE OF TROY WEIGHT. 


#4 grains (gr.) = 1 pennyweight (pwt.) 
20 pennyweights = 1 ounce (oz.) 
12 ounces = 1 pound (lb.) 


ORAL. 


How many grains in 2 pwt.? 

How many pwt. in 72 grains? 

How many oz. in 5 lb.? In 7 lb.? 

How many oz. in 100 pwt.? In 140 pwt.? 


$s & ON 


WRITTEN. 


How many pwt. in 18 lb.? 
How many gr. in 15 0z.? 
How many pwt. in 9600 gr.? 
How many lb. in 14400 pwt.? 
What part of a lb. is 1 pwt.? 

10. What is the value of 9 oz. of pure gold, if 1 oz. is 
worth $20.672? 

11. What is the value of 7 lb. of pure silver, if 1 oz. is 
worth $1.388? 

12. A gold dollar weighs 1 pwt. What is the value of 
a bag of gold dollars weighing 5 |b.? 

13, What is the value of 1 gr. of pure gold? 

See example 10. 

14. What is the difference in value between 5 oz. of 
pure gold and 5 lb. of pure silver? 

204. Apothecaries’ weight is used in compounding 
prescriptions. 


las ame ep 


TABLE OF APOTHECARIES’ WEIGHT. 


20 grains (gr.) = 1 scruple (sc. or 9.) 
3 scruples —~ oy cram (dt. or 3:.) 
8 drams = 1 ounce (oz. or 3.) 


12 ounces = 1 pound (lb. or bb.) 


174 ELEMENTARY ARITHMETIC. 


ORAL. 


J. How many gr. in 5 sc.? In 8sc.? In 34 sc.? 

2, How many dr. in 18 sce. In 60 sc.? In 90 sce.? 
3. How many oz. in 96 dr.? In 40 dr.? In 60 dr.? 
4. How many oz. in 20 lb.? In 30 lb.? In 40 lb.? 
0. How many Ib. in 24 0z.? In 36 0z.? In 72 02z.? 


WRITTEN. 


6. How many gr. in 8 scruples? 
7. How many sc. in 17 lb.? 
5. How many drams in 7200 gr.? 
9. How many ounces in 200 prescriptions, each weigh- 
ing 3 grains? 
10, How many Ib. in 28800 gr.? 


205. MEASURES OF TIME. 


TABLE OF TIME. 


60 seconds (sec.) = 1 minute (m.) 

60 minutes = 1 hour (h.) 

24 hours = .1 day (d.) 

7 days = 1 week (wk.) 

30 days = 1 month (mo.) 
* 365 days = 

12 months =+;1 year (yr.) 

52 weeks 1 day = 

100 years = 1 century (C.) 


NoTeE I.—A month of 30 days is called a business month. The actual number of 
days in the different months of the year varies from 28 to 31, as may be seen in the 
table below. 

NOTE II.—Leap year has 366 days. 

NOTE III.—Every year whose number is divisible by 4 is a leap year, unless its 
number ends with a double cipher, in which case its number must be divisible 
by 400. Thus, 1872, 1876, etc., are leap years; 1875, 1878, etc., are not leap years, 
1600 and 2000 are leap years; 1800 and 1900 are not leap years. 


The year is divided into seasons and months as follows: 


DENOMINATE NUMBERS. 175 


Abbrevi- No. of 


Seasons. Months. ations. Days. 
1. January. Jan. 31. 

es 2. February. Feb. 28. In leap year 29. 
3. March. jul sais We 
Spring 4, April. Apr. 30. 
5. May. — 31. 
6. June. — 30. 
Summer : 7. July. —- 31. 
8. August. Pet ta Ls 
9. September. Sept. 30. 
Autumn }10 October. iste sols 
11. November. Nov. 30. 
Vinter 12. December. Dec. dl. 


ORAL. 


J. How many minutes in a quarter of an hour? In 3 

quarters? In half an hour? 
How many minutes in 2 hours? In 5d hours? 
How many days in 7 weeks? In 10 weeks? 
How many weeks in 21 days? In 35 days? 
How many years in 60 months? 
How many months in 4 years? 
How many years in a quarter of a century? 
How many weeks in February, 1875? 
How many days in February, 1876? 
How many days from March 10 to May 30? 

11, What o’clock is 6 hours after 9 o’clock A.M.? 

12. What day of the week is 2 hours before 1 o’clock 
A.M. on Wednesday? 


= 
SSS NLS. Sv SoS 


WRITTEN. 


13. Tow many seconds in 5 hours? 
14. How many days in 1440 hours? 
15. How many hours in 5400 minutes? 
16. How many weeks in 1696 days? 


176 © HLEMENTARY ARITHMETIC. 


17. How many years in 25 centuries? 

18. How many days in 10 years, including 3 leap years? 
19. How many days in the last six months of the year? 
20, What date is 90 days after May 13? 

21. What date is 3 yr. 5 mo. 20 d. after April 1, 1875? 


206. PAPER. 


P+ See Le eee 
20 quires = 1 ream. 
LO Yea see 


ORAL. 


How many sheets in 2 quires? In 4 quires? 

How many sheets in $ quire? In ? quire? 

How many quires in 72 sheets? In 240 sheets? 
How many quires in 3 reams? In 5 reams? 

How many reams in 4 bales? In 6 bales? 

How many bales in 30 reams? In 60 reams? 

How many sheets in 1 ream? In } ream? 

How many quires in 4 ream? How many sheets in 


DON ek rk em eas Se ae 


% ream? 


WRITTEN. 


9° How many quires in 14400 sheets? 
10. How many reams in 600 quires? 
11, How many sheets in 1 bale? 
12. How many bales in 540 reams? 
13. How many quires in 25 bales? 


207. MISCELLANEOUS. 
12 units = 1 dozen (doz.) 
12 dozen = 1 gross. 
12 gross = 1 great gross. 


20 units = +1 score: 


DENOMINATE NUMBERS. gar a 


ORAL. 
J. How many units in 5 dozen? In 34 dozen?s 
2. How many dozen in 72 units? In 108 units? 
3. How many units in a gross? 
4. How many dozen in a great gross? 
}. What part of a gross are 72 units? 
6. How many units in 3 score? In 8 score? 
7. How many score in 100 units? In 140 units? 
WRITTEN. 
&. How many units in 1 great gross? 
9. How many units in 35 score? 


JO. How many gross in 720 dozen? 

ZI. How many dozen in 527 great gross? 
12, How many dozen in 172800 units? 
13, What part of a great gross is a unit? 
14. What part of a score is a dozen? 

15. What part of a gross is a score? 

16. How many score in 10 dozen? 

17. How many gross in 36 score? 


REVIEW QUESTIONS. 


Into what two classes may numbers be divided in regard to the 
unit? Define abstract number. Define denominate number. Define 
simple denominate number. Define compound denominate number. 
What is a solid? A surface? A line? For what are linear measures 
used? Repeat the table. What is a plane surface? What is a plane 
figure? A quadrilateral? What is an angle? A right angle? A 
rectangle? A square? How is surface measured? Repeat the table. 
How may area or surface be considered? What is acircle? <A cir- 
cumference? An arc? A degree? A diameter? A radius? By 
what is an angle measured? Repeat the table of angular measure. 
What is a sextant? A sign? What is a solid? A cube? How is 
solidity or volume measured? Of what may volume be considered 
the product? Repeat the table of cubic measure. Of liquid measure. 
How many cubic inches in a gallon? Repeat table of dry measure. 


178 ELEMENTARY ARITHMETIC. 


How many cubic inches in a bushel? How is avoirdupois weight 
used? Repeat the table. Where it the long ton used? How does it 
differ from the common ton? How is Troy weight used? Repeat the 
table. How is apothecaries’ weight used? Repeat the table. Repeat 
the table of time. How many days in leap year? How may leap 
year be known? Repeat the table of paper. The miscellaneous table. 


REDUCTION OF DENOMINATEH NUMBERS. 


208. The Reduction of a denominate number is a 
change of its expression without a change of its value. 

Thus, 3 feet may be changed to {’yard; 1 foot 6 inches may be 
changed to 18 inches, or to % yard. 


209. Reduction Descending is the change from 
a higher to a lower denomination. 


1 gal.=4 qts. 


210. Reduction Ascending is the change from 
a lower to a higher denomination. 


4 qts.=1 gal. 
REDUCTION DESCENDING. 


211. 1. How many inches in 5 yd. 2 ft. 8 in.? 


PROCESS. 
5 yd. 2 ft. 8 in. 
3 OUTLINE OF ANALysIs.—Reduce the 5 yd. 
to ft.; add the 2 ft.; reduce the ft. to inches; 
add the 8 inches. 

AwNatysis.— 1. In 1 yd. there are 3 ft.: in 
1 ft 5 yds. there are 5 times 3 ft., or 15 ft. 
ik 2. 15 ft+2 f.=17 ft. 
wake 3. In 1 ft. there are 12 inches; in 17 ft 

there are 17 times 12 inches, or 204 inches. 


se a Aeon 8 ie ol Stine aes 


— 


212 in. 


15 ft. 
2 ft. 


DENOMINATE NUMBERS. 179 


Rute for Reduction Descending. 


a 


Use as factors the number of the highest denom- 


ination given, and the value of one of that denomination 
in units of the next lower denomination. The product will 
be units of the lower denomination. | 


te 


T° this product add the given number, if any, of that 


denomination. 

Il. Make this sum a factor of another product (as in I), 
and so proceed, until the numbers are reduced to the required 
denomination. 


10. 
ih 
12. 
138. 
14. 
18. 
16. 
17. 


How many seconds in 1 h. 15 m. 30 sec.? 

How many sq. ft. in 25 sq. yd. 7 sq. feet? 

Change 4 bu. 3 pk. 2 qt. 1 pt. to pints. 

Reduce 2 T. 15 cwt. 25 |b. to |b. 

Reduce 3 bbl. 20 gal. to qt. 

How many cu. ft. in 43 cords? 

Change 3° 30’ 40” to seconds of time. 

How many sheets in 2 reams, 10 quires, 12 sheets? 
How many units in 1 great gross, 2 gross, 5 dozen? 
Reduee 2 weeks to minutes. 

Reduce a leap year to hours. 

How many years in # of a century? 

Reduce 18 cwt. 60 Ib. 12 02. to oz. 

Reduce 1 T. 80 Ib. to oz. 

How many grains in 5 lb. 3 oz. 4 pwt. of gold? 
How many scruples in 10 Ib. 7 dr. 2 sc., apothe- 


caries’ weight? 


18. 
19, 
20. 
21. 


24, 
Db. 


Reduce 2 bu. 6 qt. to pints. 

Reduce 5 hhd. to qt. 

Reduce 8 bbl. 16 gal. 2 qt. 1 pt. 1 gi. to gi. 

How many cu. in. in 8 cu. yd. 4 cu. ft. 100 cu. in.? 
How many seconds in 1 cirf. 60°? 

How many sq. rd. in 3 sq. m. 320 A. 40 sq. rd.? 
Reduce 1 m. 80 rd. 4 yd. to ft. 

How many miles in the equator? 


See Note to table of angular measure. 


180 ELEMENTARY ARITHMETIC. 


26. How many cu. in. in 4 gal.? 
See Note to table of liquid measure. 
27. How many cu. in. in 25 gal.? 
28. How many cu. in. in 3 bu.? 
See Note to table of dry measure. 

29. How many cu. in. in 40 bu.? 
30. How many cu. in. in 1 pk.? 


REpucTION ASCENDING. 


212%. 1. Reduce 101 pints, dry measure, to integers of 
higher denominations. 


PROCESS. OUTLINE OF ANALYsIs.—Reduce 
2)101 pt. pints to quarts; quarts to pecks; 
pecks to bushels. 

8) 50 qu. 1 pt. ANALYsis.—l. There are 2 pts. in 

1 qt.; in 101 pints there are as many 

4) 6 pk. 2 qt. quarts as 2 pints are contained times 
eee bs in 101 pints; +34 gt.=50 gt. 1 pt. 

1 bu. 2 pk. 2. There are 8 qt. in 1 pk.; in 50- 


qt. there are as many pecks as 8 

101 pt.=1 bu. 2 pk. 2qt.1 pt. Ans. quarts are contained times in 50 
quarts; %.° qt.=6 pk. 2 qt. 

3. There are 4 pk. in 1 bu.; in 6 pk. there are as many bushels as 
4 pecks are contained times in 6 pecks; $ bu.=1 bu. 2 pk. 


Rute for Reduction Ascending. 

I. Divide the number given by the number of its denom- 
ination which equals a unit of the next higher denomination. 
The quotient will be the number of that denomination. 

Il. Divide this quotient by the number of its denomination 
which equals a unit of the next higher denomination. So 
proceed. The last quotient, with the several remainders, will 
be the result sought. 


NOTE.—Reduction Ascending is the converse of Reduction Descending. One 
is proof of the other. 


2. How many ft. in 720 inches? 
3. Reduce 1280 rods to miles. 


DENOMINATE NUMBERS. 181 


4. Reduce 3729 sq. in. to sq. yd. 

5d. Reduce 9200 sq. rd. to sq. miles. 

6. Reduce 10900’ to cirf. 

7. Reduce 48960 cu. in. to cords. 

S. Reduce 6400 gi. to gal. 

9. Reduce 1200 pt. to bu. 
10. Reduce 18246 oz. ay. to integers of higher denom. 
11. Reduce 5600 gr. Troy to lb. Troy. 
12. Reduce 3000 gr. apoth. to lb. apoth. 
13. Reduce 7200 m. to weeks. 
14. Reduce 148976 h. to years. 
15. Reduce 6249 sheets to bales. 
1G. Reduce 5897 ft. to miles. 


SOLUTION. 


5897 ft.+3=1965 yd. 2 ft. 

1965 yd.+-54=3930+11=357 rd. 14 yd. 

357 rd.+-320=1 m. 37 rd. 

5897 ft.=1 m. 37 rd. 13 yd. 2 ft. 

But + yd.=1 ft. 6 in.; 1 ft. 6 in.+2 ft.=3 ft. 6 in.=1 yd. 0 ft. 6 in. 
1 yd.1 yd. 0 ft. 6 in.=2 yd. 0 ft. 6 in. 

5897 ft.=1 m. 37 rd. 2 yd. 0 ft.6in. Ans. 


17. Reduce 11221 ft. to miles. 
18. Reduce 625 sq. ft. to higher denominations. 


MISCELLANEOUS PROBLEMS. 


ORAL. 


1. Find cost of 83 yds. of ribbon, worth 20 cents per yd. 


Anatysis I.— 84 yd. of ribbon cost 84 times the cost of 1 yd. 
1 yd. cost 20 cents, 84 yd. cost 84 times 20 cents, or $1.70. 

Anatysis II.—Since 1 yd. cost 20 cents, 84 yd. cost 84 times 20 
cents, or $1.70. 

Awnatysis III.— 8} yd. of ribbon at 20 cents per yd. cost 84 times 
20 cents, which is $1.70. ¥ 


NoTe.—The teacher should direct the pupil which analysis to use. Analyses 
should be used till the pupil understands the logic of the example; then results only 


182 ELEMENTARY ARITHMETIC. 


should be demanded, and these as quickly as possible. Teachers should improvise 
oral examples until accuracy and rapidity are secured. For other forms of analysis, 
see pages 72, 81 and 89. 


2. Find cost of 50 feet of rope, at 4 cents per foot. 
3. Find cost of 8 yd. calico, at 124 cents per yd. 
4. Find cost of 1 rod of fence, if 7 rods cost $175. 
Awnatysis I.— 1 rod costs one seventh of the cost of 7 rods. 7 
rods cost $175; 1 rod costs one seventh of $175, or $25. 
Awnatysis I].—Since 7 rd. cost $175, 1 rod costs one seventh of 
$175, or $25. 
Awnatysis IIJ.— 1 rod costs one seventh of $175, the cost of 7 
rods, or $25. 
5. Find cost of 1 ft. of wire, if 60 feet cost $1.20. 
6. Find cost of 1 sq. yd. of carpet, if 9 sq. yd. cost 


7. Find cost of 40 acres of land, at $2.50 per acre. 
S. Find cost of 7 gal. vinegar, at 123¢ per gal. 
9. Find cost of 5 bbl. ale, at $94 per bbl. 
10. Find cost of 30 bu. wheat, at $1.50 per bu. 
11. Find cost of 50 Ib. sugar, at 12¢ per lb. 
12, What is the value of 12 T. coal, at $8.50 per T.? 
IS. What is the value of 10 oz. standard silver, worth 
$1.25 per oz.? 


14.. What is the cost of 8 lb. coffee, if 5 Ib. cost $1.50? 

ANALYsIs.—Since 5 1b. cost $1.50, 1 Ib. cost + of $1.50, which is 
$.30; and 8 Ib. cost 8 times $.30, or $2.40. 

15. What is the cost of 3 lb. sugar, if 7 Ib. cost 63 
cents? — 

16. What is the cost of 12 bu. oats, if 5 bu. cost $2? 

17. If 4 yd. of cloth cost $16, what is the cost of 9 yd. 
at the same rate? 

18. If 20 gal. wine are worth $10, what is the value of 
17 gal. of the same article? 

19. Tf 15 cords of wood are worth $45, what is the value 
of 11 cords? . 

20. Cost of 5 cwt. of flour, if 12 cwt. cost $48? 


21. 


DENOMINATE NUMBERS. 183 


Cost of 40 lb. nails, if 5 Ib. cost 25 cents? 


SuGGESTION.— 40 is 8 times 5. 


23. 


24. 


Cost of 24 T. coal, if 6 T. cost $42? 
Cost of 36 bu. oats, if 12 bu. cost $3.60? 


Cost of 2 pk. of meal, at 6¢ per qt.? 


Ana.ysis 1—At 6¢ per qt., 1 pk. costs 48¢, and 2 pk. cost 96¢. 


Anatysis IJ.—Since there are 8 qt. in 1 pk., 1 pk. costs 8 times 
as much as 1 qt., that is, 8 times 6¢, or 48¢; and 2 pk. cost 2 times 
48¢. or 96¢. 

Anatysis III.—In 2 pk. there are 16 qt.; since 1 qt. costs 6¢, 16 
qt. cost 16 times 6¢, or 96¢; that is, 2 pk. cost 96¢. 


25. 
26. 
27. 
28. 
29. 
30. 
Ol, 
32. 
34. 


What is the cost of 3 bu. corn, at 10¢ per pk.? 
What is the cost of 4 bu. wheat, at 20¢ per pk.? 
What is the cost of 5 pk. of potatoes, at 3¢ per qt.? 
Cost of 2 lb. cinnamon, at 20¢ per oz.? 

Cost of 7 T. iron, at $3 per cwt.? 

Cost of 9 yd. cord, at 3¢ per ft.? 

Cost of 1 hhd. wine, at $2 per gal.? 

Cost of 2 bbl. vinegar, at 40¢ per gal.? 

Cost of 44 gal. milk, at 5¢ per qt.? 

Cost of 34 Ib. honey, at 2¢ per oz.? 

Cost of 3 oz. sago, at 48¢ per lb.? 


ANALYsIS.— 1 oz. costs 7g of 48¢, or 3¢; 3 oz. cost 3 times 3¢, 


or 9¢. 


36. 
37. 
38. 
39. 

$60? 
40. 

$960? 


Al, 
42, 


Cost of 3 quires of paper, at $4 per ream? 

Cost of half a dozen thimbles, worth 60¢ a gross? 
Value of 5 oz. of gold, worth $160 per lb. Troy? 
Value of ? of an acre of land, if 3 acres are worth 


Value of 80 acres of land, if a sq. mile is worth 


WRITTEN. 


What is the value of 120 T. coal, worth $8.25 per T.? 
What are 73 cords of wood worth, at $2.50 per cd.? 


184 


48. 
Ad. 
Mb. 
46. 
Li. 
48. 
49. 
50. 


sy 
$102? 
52. 


ELEMENTARY ARITHMETIC. 


Cost of 224 bbl. flour, at $6.75 per bbl.? 

Find cost of 40 gal. niolasses, worth $.373 per gal. 
Value of 160 acres of land at $15.75 per A.? 

Value of 36 oz. pure gold, worth $20.672 per 0z.? 
Value of 45 oz. standard gold, worth $18.605 per oz.? 
Cost of 18 cwt. of tobacco, worth $75 per cwt.? 
Cost of 325 bu. rye, worth 623¢ per bu.? 

Cost of 70 perch of stone, at $45 per perch? 


What is the cost of 360 bbl. flour, if 17 bbl. cost 


What is the cost of 37 cords of wood, if 9 cords 


cost $29.25? 


dS. 
54. 
dd. 
56. 
d7. 
dS, 
59. 
60. 


Gis 
62. 
63. 
64. 
65. 
66. 
per sq. 
68. 
69. 
70. 


fk 


Cost of 532 bu. wheat, if 13 bu. cost $16.25? 

Cost of 184 T. coal, if 5 tons cost $25? 

Cost of 30 acres of land, if 54 acres cost $33? 

Cost of 52 reams of paper, if 26 reams cost $78? 
Cost of 54 cu. yd. of earth, if 3 cu. yd. cost $2.25? 
Value of 8% yd. cloth, if 214 yd. cost $42.50? 
Value of 24 miles of wire, if 24 miles cost $42? 
Value of 57 acres of land, if 160 A. are worth $400? 


Find cost 30 bu. of beans, at $1.125 per pk. 

Find cost of 21 gal. cider, at 44¢ per qt. 

Find cost of 3 T. nails, at 3¢ per lb. 

Find cost of 33 sq. rd. of land, at $640 per A. 

Find cost of 320 sy. yd. of oil-cloth, at 16¢ per sq. ft. 
What is the value of 1} sq. miles of land, at $.30 
rd.? 

Value of 17 quires of paper, worth $2.40 per ream? 
Value of 5 acres of land, worth $.50 per sq. ft.? 
Value of 3 cu. yd. of stone, worth $6.40 per cord? 
Cost of 5 hhd. vinegar, worth 8¢ per qt.? 


Find cost of 2 yd. 1 ft. 6 in. of chain, at 50¢ per yd. 


SuGG@EsTion.—Reduce the ft. and in. to fractions of a yd. 


DENOMINATE NUMBERS. 185 


72. Find cost of 4 A. 40 sq. rd. of land worth $60 per A. 

73. Find cost of 6 cd. 32 cu. ft. of wood at $4 per cd. 

74. Find cost of 20 perch 84 cu. ft. of stone, at $3.30 
per perch. 

75. Find cost of 400 cu. yd. 13 cu. ft. 864 cu. in. of 
masonry, at $1.75 per cu. yd. 

76. Value of 15 bbl. 15 gal. 3 qt. spirits, worth $150 
per bbl.? 

77. Cost of 85 bu. 3 pk. corn, at $.50 per bu.? 

78. Cost of 32 T. 2 cwt. 50 |b. of iron ore, worth $12 
per T.? 

79. Value of 7 oz. 15 pwt. of gold, worth $20.50 per oz.? 

SO. Cost of 4 gr. gross 8 gross buttons, at $2.50 per 
er. gross? 


ADDITION AND SUBTRACTION. 


213. 1. Find the sum of 3 yd. 1 ft. 7 in.; 2 ft. 8 in.; 
1 yd. 1 ft. 10 in. 


PROCESS. ANALYsISs.—1l. Write the numbers to be added so 
yd, ft. in, that units of the same denomination shall stand in 
3 1 7 the same column. 
2 8 2. Find the sum of the numbers in the column 
1 ae sith of the lowest denomination (inches), which is 25 
oe Se inches=2 ft. 1 in. Write the 1 in. under the column 
6 0 7 of in.; add the 2 ft. to the column of ft. 

3. Find the sum of the column of ft., which is 6 
ft.=2 yd. 0 ft. Write the 0 under the column of ft.; add the 2 yd. to 
the column of yd. 

4. Find the sum of the column of yd., which is 6 yd. 


NOTE.—By comparing the analysis of addition of simple numbers, page 41, it 
will be seen that the same principles underlie the addition of both simple and com- 
pound numbers. In the addition of simple numbers, the sum of each column is 
always divided by 10, the remainder written, and the quotient added to the next 
higher column. In the addition of compound numbers, the sum of each column is 
divided by the number of units which equal one of the next higher denomination, 
the remainder written, and the quotient added to the next higher column. In sub- 
traction, multiplication and division, the same principles are applied. 


Rute for Addition of Compound Numbers. 
I. Write the numbers to be added so that units of 
the same denomination shall stand in the same column. 


186 ELEMENTARY ARITHMETIC. 


Il. Find the sum of the numbers in the column of the 
lowest denomination. 

III. Lf this sum is less than the number required to 
equal a unit of the next higher denomination, write it 
under its own column. If the sum is greater than the num- 
ber required to equal a unit of the next higher denomina- 
tion, divide it by this number; write the remainder, if any, 
under its column, and add the quotient to the next higher 
column. 

IV. So proceed. Under the highest column write its 
sum. 


4. Add.d5 1". 8 cwt. 10 Ib. 6 oz.; 2 T: 16 cwt. 45 Ib. 8 
oz.; 7d lb. 10 oz. 

3. Add 10 bu. 1 pk. 2 qt.; 7 bu. 3 pk. 5 qt.; 20 bu. 4 qt. 

ZeeeAOdO-miol40-Tdo ydise 20m. 10rd: 92 tts bein. 
Wow ttett-in,s)1Q m= 00 Td-so.vdeoun. 

Oe AU 252-3940" (2°81 656 al 209s 5 aL eee 

6. Add’5: A. 120 'sq. rd. 7 sq. ft. 129'sq. in.3) 20 A. 40 
sq. rd. 25 sq. yd. 8 sq. ft.; 1sq.m. 50 A. 100 sq. rd. 

7. From 18 gal. 2 qt. 1 pt. 2 gi. subtract 10 gal. 1 qt. 
tte O01, 

PROCESS. _ANALYsIs.—1. Write the subtrahend under 
gal. qt. pt. gi. the minuend, units of the same denomination 
pes OE. 128. in the same column. 

10 1 1 3 2. Subtract, beginning with the lowest de- 
SR ey ae Ee RTO nomination. Since it is impossible to take 3 

oe ce cae ai. from 2 gi., take from the minuend the 1 
pt.=4 gi. and add to the 2 gi.; 2 gi+4 gi.=6 gi.; 6 gi.—3 gi.=3 
gi., which write under the column of gi. 

3. Since the 1 pt. of the minuend has been taken, 0 pt. remain, and 
it is impossible to take the 1 pt. of the subtrahend from 0 pt. of the 
minuend. Therefore, take 1 qt. from the 2 qt. of the minuend, and 
reduce it to pt.; 1 qt.=2 pt.; 2 pt.—1 pt.=1 pt., which write under 
the column of pt. 

4. Since 1 qt. has been taken from the 2 qt. of the minuend, 1 qt. 
remains. 1 qt.—1 qt.=0 qt., which write under the column of qt. 

5. 18 gal.—10 gal.=8 gal. 

Result, 8 gal. 0 qt. 1 pt. 3 gi. 


DENOMINATE NUMBERS. 187 


Rutle.—The pupil may easily deduce a rule from the 
analysis, or from an example. 


8. From 10 yr. 7 mo. 8 d. subtract 5 yr. 9 mo. 20 d. 
9. From 20 bu. 3 pk. 1 qt. 1 pt. subtract 15 bu. 2 pk. 
6 qt. 1 pt. 
10. From 1 T. take 5 ewt. 25 lb. 10 oz. 
11. From 1 cd. take 50 cu. ft. 200 cu. in. 
12. From 1 bbl. 5 gal. 3 qt. take 25 gal. 1 qt. 1 pt. 
18. From a quadrant take a sextant. 
14. From 5 lb. 8 oz. 16 pwt. take 3 lb. 9 oz. 5 pwt. 16 gr. 


MULTIPLICATION. 
214. 1. Multiply 3 pk. 7 qt. 1 pt. by 9. 


ist PROCEsS. 


pk. qt. pt. 
3 7 1 
9 


8 bu. 3. pk. 3 qt. 1 pt. Ans. 


2nD PROCESS. 


pk. qt. pt. 

3 Y 1 

9 

Products - - 27 63 9 
Quotients - — - 8 4 
Amounts” - : al 67 


8 bu. 3 pk. 3 qt. 1 pt. Ans. 


ANALysis.—1. Beginning with the lowest denomination, 9 times 
1 pt.=9 pt.=4 qt. 1 pt. 

2. 9 times 7 qt.=63 qt.; 63 qt+4 qt.—67 qt.—8 pk. 3 qt. 

3. 9 times 3 pk.=27 pk.; 27 pk.+8 pk.=35 pk.=8 bu. 3 pk. 


Novre.—In the 2nd process the work is more extended. 


188 HLEMENTARY ARITHMETIC. 


Rule.—tLet the pupils deduce a rule, and compare it 
with the rule for multiplication of simple numbers. 


Multiply 8 yd. 2 ft. 5 in. by 7. 

Multiply 13 lb. 10 oz. Av. by 20. 

Multiply 24° 15’ 28” by 18. 

Multiply 7 lb. 5 oz. 12 pwt. 8 gr. by 24. 
Multiply 129 cu. ft. 524 cu. in. by 32. 
Multiply 3 bales, 7 reams, 9 quires by 12. 
Multiply 25 cu. yd. 17 cu. ft. 34 cu. in. by 40. 
Multiply 5 T. 3 cwt. 40 lb. 8 oz. by 28. 


6 Go ND APR bo & 


I PWEDE BEES 

215. 1. Divide 7 lb. 3 oz. 11 pwt. by 4. 
Process. ANALYSIS. — 1. Begin 
5 with the highest denom- 
yi lDajeax0 OZ. aes tlaywes ination; one fourth of 7 
Ib. is 1 1b. with remainder 

HENS PE prays owt. 8 or. 
Bie 0 OZ wali Wire hie oT eee 


2. 3 1b.=86 0oz.; 36 0z.+3 0z.=39 oz.; one fourth of 39 oz. is 9 oz. 
with remainder of 3 oz. 

3. 3 02z.=60 pwt.; 60 pwt.t11 pwt.=71 pwt.; one fourth of 71 
pwt. is 17 pwt. with remainder of 3 pwt. 

4. 3 pwt.—72 er.; one fourth of 72 gr. is 18 gr. 

Rule.—1. Divide the number of the highest denom- 
ination given. The quotient is the first term of the result 
sought. — 

Il. Reduce the remainder, if any, to the next lower de- 
nomination. Hind the sum of the reduced remainder and 
the number, if any, of the same denomination of the divi- 
dend. 

Ill. Divide as before, and so proceed. 


2. Divide 16 yd. 2 ft. 10 in. by 3. 

3. Divide 1 T. 18 ewt. 50 lb. 8 oz. by 4. 
4. Divide 60° 27’ 40” by 12. 

45. Divide 8 bu. 3 pk. 7 gt. by 6. 


DENOMINATE NUMBERS. 189 


6. Divide 10 A. 81 sq. rd. by 9. 
7. Divide 48 cu. yd. 3 cu. ft. 190 cu. in. by 36. 


PROBLEMS. 


J. A farmer sold 3 loads of corn at 373¢ per bu.; the 
first load contained 388 bu. 2 pk., the second 40 bu. 5 qt., the 
third 29 bu. 1 pk. 3 qt. What did he receive for the three 
loads? 

2. From a cask of wine containing 40 gal., 3 gal. 1 qt. 
leaked out, and 20 gal. 2 qt. 1 pt. were drawn. How much 
remained? 

3. A merchant wished to buy 400 yd. of cloth; after 
having purchased 5 pieces, each containing 32} yd., and 10 
pieces each containing 184 yd., how many yd. must he buy? 

4. If aman can walk 3 m. 150 rd. in 1 hour, how far 
can he walk in 7 hours? 

5. What will be the size of each farm if 640 acres be 
divided equally into 6 farms? 

6. John is 18 y. 9 mo. 10 d. old; James is one fourth as 
old. How much older is John than James? 

7. A locomotive runs 180 miles in 8 hours. What is 
its average speed per hour? 

5S. How many books can be made from 1 bale of paper, 
if each sheet makes 8 leaves, and each book contains 160 
pages? 

9. A great gross of spools of cotton was sold for 5¢ per 
spool. What was the price of the lot? 

10. If 1 dose of medicine contains 2 sc. 15 gr., how 
many lb. Troy, ete., in 120 such doses? 

17. If aman can cut 1 cd. 12 cu. ft. in 1 day, what can 
he cut in three eighths of a day? 

12. From a bbl. of alcohol 5} gal. were drawn, and the 
remainder filled 72 bottles, each holding 3 pt. How much 
alcohol was there in the bbl. at first? 

15. The sun moves at the rate of 15° of longitude in 1 
hour. In what time does it move 100° 50’ 30”? 


190 ELEMENTARY ARITHMETIC. 


14. In what time does the sun move 87° 31’ 45”? 

15. Through how many degrees, minutes, and seconds 
of longitude does the sun move in 7 h. 51 m. 47 sec.? 

16. Through how many degrees, etc., does the sun move 
in 9 h. 59 m. 24 sec.? | 

17. What is the weight of a dozen silver spoons, if 
each weighs 16 pwt. 18 gr.? 

18. Three fourths of a lb. of pure silver, mixed with 1 
oz. of alloy, is made into a dozen napkin rings. What is 
the weight of each ring? 


REVIEW QUESTIONS. 


What 1s the reduction of a denominate number? How many kinds 
of reduction? Define and illustrate each. Rule for reduction des- 
cending. Rule for reduction ascending. How may each kind of 
reduction be proved? What is the difference between addition of 
simple numbers and addition of compound numbers? Give rule for 
addition of compound numbers. Give rule for subtraction of com- 
pound numbers. Rule for multiplication of compound numbers. 
Rule for division of compound numbers. 


MENSURATION OF RECTANGLES AND REOTAN- 
GULAR SOLIDS. 


For definitions and principles, see Arts. 183, 186, 196 and 198. 


ORAL. 


216. 1. How many square inches on one side of a 
slate 10 in. long, 6 inches wide? How many sq. in. on both 
sides? 

. 2. How many sq. in. on one page of a sheet of foolscap, 
12 in. long and 8 in. wide? 

3. How many sq. ft. in a blackboard 4 ft. wide and 10 
ft. long? 

4. How many sq. in. in a surface 8 in. by 3in.? 9 in. 
by 7in.? 7 in. by 4in.? 30 in. by 8 in.? 


DENOMINATE NUMBERS. 191 


5. How many sq. ft. in the ceiling of a room 15 ft. long 
and 10 ft. wide? 
6. How many sq. yd. in the floor of a room 12 ft. by 
11 ft.? 
7. How many sq. ft. in a wall 9 ft. by 15 ft.? How 
many sq. yd.? | 
S. The cover of a book is 9 in. long, and contains 63 
sq. in. How wide is it? 
9. The top of a desk contains 15 sq. ft.; it is 3 feet 
wide. How long is it? 
10. How many sq. in. in the entire surface of a brick? 
A brick is 8 in. by 4 in. by 2 in. 
11, How many sq. in. in the entire surface of a cube 3 
in. on each edge? 
12. What part of a sq. yd. is a surface 2 ft. by 2 ft.? 
13. What part of a sq. ft. is a surface 3 in. by 12 in.? 
14. What part of a sq. yd. is a surface 1 yd. long and $ 
yd. wide? 
15. What part of a sq. ft. is a surface 9 in. by 8 in.? 
16. How many cu. in. in a block 5 in. long, 3 in. wide, 2 
in. thick? 
17. How many cu. in. in a brick? 
18. How many cu. in. in a stone 10 in. by 8 in. by 4 in.? 
19. How many cu. yd. in a block of marble 9 ft. long, 3 
ft. wide, 2 ft. thick? 
20. How many cu. ft. in a log 1 foot square and 100 ft. 
long? 
21. How many cu. ft. in a box 3 ft. by 4 ft. by 33 ft.? 
22. How many cu. ft. in a wall 20 ft. long, 5 ft. high, 2 
ft. thick? 
23. How many cu. ft. in a solid 3} ft. by 4 ft. by 14 ft.? 
24. How many cu. ft. in a solid 8 ft. by 24 ft. by 54 ft.? 
25. How many cu. ft. in a solid 2 yd. long, 3 ft. wide, 
14 ft. thick? 
26. A box contains 64 cu. in.; it is 4 in. wide, and 2 in. 
thick. How long is it? 


192 ELEMENTARY ARITHMETIC. 


27. A solid contains 144 cu. in.; it is 2 in. thick, 6 in. 
wide. How long is it? 


WRITTEN. 


28. How many sq. ft. ina surface 172 ft. long, 805 ft. wide? 

29. How many sq. yd. ina surface 324 ft. long, 81 ft. wide? 

30. How many sq. rd. in a surface 320 rd. by 1280 rd.? 
How many acres? How many square miles? 

31. How many sq. ft. in the walls of a room 21 ft. long, 
12 ft. wide, 9 ft. high? 

Find the product of the perimeter of the room, or the entire 
length of its four walls, and its height; thus: (21 ft.x2)+(12 ft.x 2) 
=66 ft., the perimeter of the room; 66X9= the entire surface of the 
four walls. 

32. How many sq. ft. in the four walls of a room 24 ft. 
long, 18 ft. wide, 12 ft. high? How many sq. yd.? 

33. Find the cost of plastering the walls and ceiling of 
the room described in Ex. 31, at 623¢ per sq. yd. 

34. Find the cost of plastering the walls and ceiling of 
the room described in Ex. 32, at 3743¢ per sq. yd. 

35. How many sq. yd. in floor, walls, and ceiling of a 
room 48 ft. long, 36 ft. wide, 15 ft. high? 

36. Cost of laying a sidewalk 200 ft. long, 14 ft. wide, 
at 8¢ per sq. ft.? 

37. How many sq. yd. in a floor 18 ft. long, 12 ft. wide? 
How many sq. yd. of carpeting necessary to cover the floor? 

38. How many yd. of carpeting, 1 yd. wide, necessary 
to cover a floor 12 ft. long and 9 ft. wide? 

39. Wow many yd. of carpeting, + yd. wide, necessary 
to cover the floor of a room 12 ft. long and 9 ft. wide? 

40. How many yd. of carpeting, + yd. wide, necessary 
to cover the floor of a room 12 ft. long and 9 ft. wide? 
How many yd. # yd. wide? 

41. Cost of carpeting, $ yd. wide, necessary to cover a 
floor 21 ft. long by 12 ft. wide, at $1.50 per yd.? 


Carpets are sold at a certain price per yd. zn length. A piece of 


DENOMINATE NUMBERS. 193 


carpet } yd. wide, 20 yd. long, contains 15 sq. yd.: but the cost at $2 
per yd. is 20 $2, not 15x $2. 

42. Cost, at $2.25 per yd., of carpet § yd. wide, neces’ 
sary to cover a ene 40 ft. long, 24 ft. wide? 

43, A blackboard containing 96 sq. ft. is 4 ft. wide. 
How long is it? 

44. A street 80 ft. wide contains 1 acre. How many ft. 
long is it? 

45. Wow many squares, each 6 in. on a side, can be cut 
from a sheet of paper 4 ft. square? 

46, Cost, at $2.50 per acre, of a tract of land containing 
6 sq. miles? 


47. How many cu. ft. in a wall 24 ft. long, 10 ft. high, 
2 ft. thick? 

48. How many cu. ft. in a wall 96 ft. long, 18? ft. high, 
24 ft. thick? 

49. How many cu. in. in a block 3 ft. by 5 ft. by 1% ft.? 

50. How many cu. ft. in a stone 144 in. long, 24 in. 
wide, 8 in. thick? 

51. How many cubes, each 3 in. on each edge, can be 
cut from 1 cu. ft.? 

O2, How'many cu. ft. in a pile of wood 16 ft. long, 4 ft. 
wide, 4 ft. high? 

53. How many cords in a pile of wood 24 ft. long, 4 ft. 
wide, 4 ft. high? 

d4. What is the value, at $6.75 per cord, of a pile of ~ 
wood 32 ft. long, 8 ft. wide, 4 ft. high? 

55. What is the cost of sawing, at 874¢ per cord, a pile 
of wood 12 ft. long, 4 ft. wide, 6 ft. high? 

56. How many perch of stone in a pile 99 ft. long, 24 
ft. wide, 10 ft. high? 

57, Find Ate of stone described in Ex. 56, at $3.124 
per perch? 

58. How many perch of masonry in a wall 494 ft. long, 
8 ft. high, 4 ft. thick? 

Q 


194 ELEMENTARY ARITHMETIC. 


59. Cost of building a wall of stone 24? ft. long, 7 ft. 
high, 2 ft. thick, at $4.623 per perch? 
GO. How many bricks in a wall 24 ft. long, 8 ft. high, 3 


ft. thick? 
: 24x8x38 x 1728 

Use cancellation, thus: ~—"~2"-"=, 
2x4x8 


61. How many bricks in a pile 40 ft. by 20 ft. by 12 ft.? 

62. Find the cost of brick necessary to build a wall 
80 ft. long, 60 ft. high, 23 ft. thick, at $8 per M.? 

63. Cost, at $9.50 per M., of the bricks necessary to 
build the walls of a house in the form of a rectangle, 40 ft. 
long, 32 ft. high, 24 ft. wide, the walls 2 ft. thick? 


Nore.—Take the outside measurement, which is given. Make no allowance for 
doors or windows. This will givethe mason’s measurement, whichis rather greater 
than the actual measurement. There are 21 bricks ina cubic foot, allowing for 
mortar. 


64. How many cu. in. in 40 gallons? 

65. How many cu. in. in 5 bbl.? 

66. How many cu. in. in 1 hhd.? 

67. How many cu. in. in 5 bushels? 

68. How many cu. ft. in 20 bushels? 

69, How many gal. in a box 40 in. by 24 in. by 18 in.? 

70. How many gal. in a box 3 ft. by 2 ft. by 14 ft.? 

71. How many gal. in a tank 8 ft. by 6 ft. by 4 ft. 

72. How many bushels may be contained in a bin 20 ft. 
by 8 ft. by 12 ft.? 


MENSURATION OF PARALLELOGRAMS. 


217. A Parallelogram isa plane four-sided figure, 
whose opposite sides are parallel and equal. 
A_E B A BCD is an oblique angled 


' parallelogram. 
Squares and rectangles are varieties 
F A of the parallelogram. 


218. The Base of any plane figure is the side 
upon which it is supposed to rest.  Parallelograms are 


D 


DENOMINATE NUMBERS. 195 


considered as having two bases, called wpper and lower 
bases. 
A Band C D are the bases of the parallelogram A BC D. 


219. The Altitude of a parallelogram is the perpen- 
dicular distance between its bases. 

E F is the altitude of the parallelogram A BC D. 

In rectangles, either end or side is the altitude, because the end or 
side is perpendicular to the base. 

220. The area of «@ parallelogram is equal to the 
product of its base and altitude. 

J, Find the area of a parallelogram having a base of 12 
ft., altitude of 8 ft. 

2, Find area of parallelogram having base 50 ft., alti- 
tude 25 ft. 


MENSURATION OF TRIANGLES. 


221. A Triangle is a plane figure bounded by 
three straight lines. 

A B Cis atriangle. C B is its 
base; the angle A, opposite the base, 
is its vertex; A D, the perpendicular 
distance from the vertex to the base, 
is its altitude. 


B 


)os4insnc-eaa-} > 


222. The Diagonal of a parallelogram is the straight 
line joining two opposite angles. 


E F 
K L 


ra) 
Se 
Ag 


N M 
is G 


E G is the diagonal of the rectangle E F G H. K M is the 
diagonal of the parallelogram K L M N. 

223, The diagonal of a parallelogram divides the 
parallelogram into two equal triangles. 


196 ELEMENTARY ARITHMETIC. 


The triangles E F G and EG H are equal. The triangles K L 
M and K M N are equal. 

Tne pupil may draw 
any triangle, and by 
annexing to it, in the 
manner shown, an equal 
triangle, may form a par- 
allelogram. 


SOew we ewe me ee eee eS 


224. Since the area of a parallelogram is equal to the 
product of its base and altitude; and since every parallelo- 
gram may be divided into two equal triangles having the 
same base and altitude as the parallelogram, it follows that 

The area of a triangle is one half of the product of ts 
base and altitude. 


Find the area of the following triangles: 


1. Base 6; altitude 4. 
2. Base 12; altitude 8. 
3. Base 40; altitude 15. 
4. Base 60; altitude 30. 
}. Base 85; altitude 20. 


6. The base of a triangle is 123; its altitude is 25. 
What is its area? 

7. The area of a triangle is 625; its altitude is 50. 
What is its base? 

S. The area of a triangle is 360; its base is 60. What 
is its altitude? 

9. The area of a parallelogram is 360; its base is 60. 
What is its altitude? 


REVIEW QUESTIONS. 


Define parallelogram; triangle. Define base of parallelogram; 
base of triangle. Define altitude of parallelogram. Define diagonal. 
How may the area of a parallelogram be found? Of a triangle? If 
a parallelogram and a triangle have same base and altitude, how do 
their areas compare ? 


. 


PERCENTAGE. 197 


pen LO Nex ks 


PERCENTAGE. 


Art. 225. Per Cent means hundredths. 

1 per cent is 74>; 6 per cent is 7%>, ete. 

226. Rate per Cent means a certain number of 
hundredths. 

In the expression 6 per cent, 6 is the rate. The rate is the numer- 
ator of a fraction of which the denominator is always 100. Instead 
‘ of the words per cent the sign % is frequently used, 

227. Rate per cent may be written as a common frac- 
tion, as a decimal fraction, or as per cent. 


Thus: 

zi7= .01=12; the last, 1%, is read, one per cent. 

Too = -06=62. 

a 7.074 or .075=7474; the last is read seven and one half per cent. 
424 — 123 or .125=1244. 

en 1.25=1252. 


7=-005=42, the last is read one half of one per cent. 


Observe that the decimal point is not used when the words per cent 
are used, or when the sign % is used, unless a fractional per cent is 
meant. Thus, 5% equals .05, not .5¢. .5% means 5 tenths of one per 
cent, and is otherwise written 47, or .004. 


Express as per cent: 


1. rive 5. Pos 9. .06. LS: PLAS 
2. ze 6. +52 repos ales LZ. 2.60 
3. Pos. ry) Bek 11. .08}. 15. .005 
4. Pay: 8. tb 12, 12k. © 16, .00376 


198 ELEMENTARY ARITHMETIC. 


Hixpress decimally: 


1. 10%. 6. 84g. 11. 100¢. 16. 74%. 

2. 15%. 7. 64%. 12. 1252. 17, 2249. 
3. Bb. 8. 52%. 18 11314. > 2 Sama 
4, 64 9. 124%. 14. 300%. 19. 13744. 
5. 2% 10. 40%. 15. 875¢. 20. 10634. 


228. The Base is the number of which the per cent 
is computed. 
229. The Percentage is that percent of the Base 
which is indicated by the Rate. 
7. A man having 500 bushels of wheat, sold 8% of it. 
How many bushels did he sell? 
PROCESS. SoLtutron.—He sold .08 of 500 bushels, 
500 x .08 =40. which is 40 bushels. 
In this example 500 bushels is the Base; 8% 
is the Rate per cent ; 40 bushels is the Percentage. 


Observe that this is but a simple problem in multiplication of- 


decimals. 

230. The Base is the MWultiplicand.. 

The Rate per cent is the Multiplier. 

The Percentage is the Product. 

231. As in simple multiplication, if any two of these 
terms are given, the other may be found, since the Base 
and Rate are factors of the Percentage. 

2, A man having 500 bushels wheat sold 40 bushels. 
What per cent. of his wheat did he sell? 


PROCESS. SoLuTion.—An inspection of example 1 will 
40-+500=.08. Show that 40 bushels is the product of two 
factors, the Base and the Rate. Dividing the 

product, 40, by the given factor, 500, the quotient is the Rate, 8%. 


3. A man sold 40 bushels of wheat, which was 8% of 
what wheat he had. How many bushels had he? 


PROcEss. SoLtutTion. —Comparing this example with 
40~.08=500, example 1 and 2, it will be seen that the factor 
required here is the Base. Dividing the product, 

40, by the given factor, .08, the quotient is the Base, 500. 


a ee ee ee 


: 
. 
P 
y 


PERCENTAGE. 199 


GASE 1.—Base and Rate per cent. given, to find the Percentage. 


232. Rule.—lind the product of the Base and the 
Rate expressed as a decimal. 
Study example 1. 


CASE Il.—Base and Percentage given, to find the Rate. 


233. Rule.—Vind the quotient of the Percentage 
divided by the Base. 
Study example 2. 


CASE III.—Percentage and Rate given to find the Base. 


234. Rule—Find the quotient of the Percentage 
divided by the Rate. 
Study example 3. 
ORAL. 


4.. What is 10% of 50? Of 60? Of 80? Of 100? 

0. What is 20% of 40? Of 80? Of 75? Of 60? 

6. Coffee was bought for 40¢ and sold at a gain of 
20%. What was the gain? 

7. A knife that cost $1 was sold at an advance of 50%. 
What was the gain? 

5. A book costing $.60 was sold at a loss of 30%. What 
was the loss? 

9, A farmer having 200 sheep lost 40% of them. How 
many did he lose? 

1O. John had 80 marbles, and sold 25% of them to James. 

How many had he remaining’? 


11. What per cent is 8 of 16? 16 of 8? 

12. What per cent is 5 of 25? 25 of 5? 

13. What per cent is 80 of 20? 20 of 80? 

14. 15 is what per cent of 60? 60 of 15? 

15. Mary having 20 peaches, gave 10 to her playmates. 
What per cent of her peaches did she give to them. 


200 ELEMENTARY ARITHMETIC. 


16, A regiment of 1000 men lost 50 men in battle. 
What per cent of its number did it lose? 

17. Isold my gold pen, which cost $2, for $1.50. What 
per cent did I lose? 

18. Julia’s watch is worth $50; Annie’s is worth $75. 
What per cent of the value of Annie’s watch is the value 
of Julia’s? 

19, What per cent of the value of Julia’s watch is the 
value of Annie’s? 

20, John bought a sled for $2, and lost it. What per 
cent of his investment did he lose? 

21. Mr. Roberts invested $2000 in business, and gained 
10% of his original investment every year. How many 
dollars did he gain in 1 year? In 5 years? . 

22. Mr. Tracy gained every year 20% of an investment. 
In how many years will he gain 40%? In how many years 
will he double his capital? 


23. Percentage 20; rate per cent 10. Find base. 

24. Percentage 30; rate per cent 25. Find base. 

25. Percentage 60; rate per cent 6. Find base. 

26. Percentage 56; rate per cent 8. Find base. 

27. $20 is 20% of John’s money. How much has he? 

28. $50 is 124% of the value of a horse. What is his 
value? 

29. A book was sold at a gain of $1. 50, which was 30% 
of its cost. What was its cost? 

30. A man traveled 40 miles by stage, which was 20% of 
his whole journey. How many miles in the journey? 

#35. When the rate per cent is an aliquot part of 100, 
the rules given above need not be followed. 

31. John had $50, and lost 10% of it. How many dol- 
lars did he lose? 

Sotution.— 10¢=745=45- Johmiost 3/5 of his money, that is, $5. 

32, James having 24 books, sold 3Y46 of them. How 
many did he sell? 


PERCENTAGE. 


SoLutTion.— 3874¢= ot. James sold $ of his books, that is, 


) books. 


33. Mary is 15 years old; Jane is 10 years old. What 


per cent of Mary’s age is Jane’s age? 


Soturion.—Jane’s age is {% or $ of M 
4. Henry gave his sister 3 app 
what he had. How many had he? 


SoLution.— 25¢4=—79,;=1. 3 was } 


whole number was 4 times 3 apples, or 12 
TABLE. 
1%=;1,. 
ee= 750 =F" 
42740 +25: 
S675 =a: 
10% =F 5 =75- 
20% =Fy'5 =4 
20% = 5, =4 
50% = foo =? 


ary? eas, O96 
ary’s age. #=7/3>= 


les, which were 25% of 


of Henry’s apples 


apples. 

84 =fh5 i eS 
125% i io =4. 
374% = 754 =8. 
6244 = hb =8- 
874¢— 88 —7, 
163%= Teh ay 


66% __ 2 
663%=7) 9s 


Nore.—The pupil should be made perfectly fam 


iliar with the above table. 


34). Find 25% of 80; of 100; of 400; of 50. 
36. Find 334% of 60; of 45; of 300; of 75. 


37. Find 124% of 40; of 160; of 
38. Find 374% of 80; of 320; of 
39. What per cent of 60 is 40? 
40. 5 is 20% of what number? 

41. 12 is 662% of what number? 
42. 14 is 873% of what number? 
43. 15 is 162% of what number? 


64; of 96. 
400; of 640. 
Of 80 is 60? 


Find WRITTEN. 
44. 16% of 200. 48. 5% of 1285. 
45. 13% of 400. 49. 9% of 840. 
46. 2% of 395. 50. 125% of 450. 
47. 38% of 750. 51. 230% of 620. 


g * 


202 ELEMENTARY ARITHMETIC. 


52. Aman sowed in oats 30% of his farm of 160 acres. 
How many acres did he sow in oats? 

53. Mr. Roberts is 80 years old; he lived 40% of his life 
in England. How many years did he live in England? 

54. 24% of a flock of sheep numbering 2800 was sold. 
How many sheep were sold? 

55. A man bought a horse for $200, and sold him so 
as to gain 10% of his cost. What did he gain? 


Note.—The cost is always the BASE. The GAIN or LOSS is the PERCENTAGE. 


06. A house was bought for $4000. It is now valued at 
125% of its cost. What is its value? 

O7. A lawyer receives 8% of all money that he collects. 
What is his fee on a collection of $500? 

OS. A merchant insures his goods for $5000, paying 
4%. What does he pay? 

SuGGEsTION.—First find 14% of $5000, by dividing by .01; then take 
# of this quotient. 


NOTE.—The money paid to insure property is called premiwm. 


59. What is the premium paid for insuring a house 
worth $8000, for one half its value, at 144? 


60, A merchant insured his store for $2000, at. 1345 


and his stock of goods for $50000, at 3%. Find his total 
premium. 

Ol. A ship was bought for $18000; repaired at an 
expense of $4000, and sold at a gain of 9% on total cost. 
What was the gain? 


Find the rate: 


62. Base 400; percentage 32. 
63. Base 900; percentage 630. 
64. Base 846; percentage 282. 
65. Base 725; percentage 125. 


66. John having $60, gave his brother $12. What per 
cent of his money did he give to his brother? 


ee 


PERCENTAGE. 203 


67. A house valued at $2000 was damaged to the extent 
of $800. What was the per cent of the damage? 

6S. A watch was bought for $80, and sold for $100. 
What was the gain per cent? 

69, A grocer buys sugar at 8 cents, and sells it at 9 
cents. What is his gain %? 

70. When a man sells for 10 cents what has cost him 
12 cents, what is his loss #? 

71. A vessel which cost $40000, was sold for $35000. 
What was the loss %? 

72. A collector charged $72 for collecting $900. What 
was his charge %? 

ry oO 


73. A house was insured for $8500. The premium was 
$127.50. Find the rate. 


Find the base: 


74. Percentage 42; rate 6%. 
75. Percentage 75; rate 5%. 
76. Percentage 91; rate 7%. 
77. Percentage 873; rate 1244. : 


7S. A man lost $50 which was 10% of his money. How 
much had he at first? 

79, Mary is 8 years old; her age is 20% of her mother’s 
age. How old is her mother? 

SO. A merchant sold $12000 worth of goods in Decem- 
ber, which was 15% of what he sold during the entire year. 
What were his annual sales? 

S/. The gain on the sale of a house was $200, which 
was 123% of its cost. What was its cost? 

82, The premium for insuring a house was $60; the rate 
was 13% For what was the house insured? 

S83. The taxes on a certain lot are $30 per year; the 
levy is 15 mills on the dollar. What is the assessed value 
of the lot? 


204 HLEMENTARY ARITHMETIC. 


S4. The loss on the sale of a horse was $50, which was 
40% of his cost. What was his cost? 


REVIEW QUESTIONS. 


What is meant by per cent? By rate per cent? In what way may 
per cent be expressed? Explain the use and non-use of the decimal 
point in writing rate per cent? Define base. Define percentage. To 
what term in multiplication does each term in percentage correspond ? 
How may each term be found? Which terms in percentage are 
factors? State and explain Case I. Case II. Case III. To which 
term in percentage does the cost of an article correspond? The gain 
or loss? Define premium. 


INTEREST. 


236. Interest is the compensation or payment due 
one party (the lender) from another party (the borrower) 
for the use of money. 

237. The Principal is the sum of money for whose 
use a compensation is made. 

238. The Rate of Interest is a certain rate per 
cent, which the interest is of the principal, for a specified 
time, usually one year. 

239, The Amount is the sum of the principal and 
interest. 


ORAL. 


1. What is the interest of $100 for 1 year, at 6%? 
SoLtutron.—The interest is 6% of $100, that is, $6. 
2, What is the interest of $40 for 3 years, at 10%? 


Sotution I.—The interest of $40 for one year, at 10% interest, is 
$4; for 3 years it is 3 times @4 that is, $12. 

Sonurton II.—The interest of $1 at 10% for one year is 10 cents; 
for three years it is 80 cents. The interest on $40 for 3 years, at 102, is 
40 times 30 cents, that is, $12. 


PERCENTAGE. 205 


What is the interest of 


PH So N_S3 Sc Se 


$80 for 1 year at 8%? Ll. $7.50 for 4 years at 5%? 
$50 for 2 years at 4%? 12, $12.50 for 2 years at 82? 
$200 for 4 yearsat 10%? 15. $3.33 for 3 years at 10%? 
$900 for 3 years at 72? 14. $50 for 2 years at 732? 
$300 for 5 years at 9%? 15. $30 for 3 years at 53%? 
75 for 4 years at5%? 16. $1 for 10 years at 10%? 
$10 for 4 years at 62? =617. $3 for 8 years at 125%? 
$3.50 for 2 yearsat 10%? 18. $5 for 4 years at 64%? 


19. What is the interest of $30 for 1 year 4 months 


at 9%? 


SoLtution.—The interest of $30 for 1 year at 9% is $2.70; for 4 
mos. (4 of a year) the interest is 4 of $2.70, or 90 cents. 
$2,70-+.90—$3.60. 
20, What is the interest of $60 for 1 yr. 3 mos. at 6%? 
21. What is the interest of $400 for 2 yrs. 6 mos. at 10%? 
22, What is the interest of $900 for 3 yrs. 2 mos. at 6%? 
23. What is the interest of $150 for 1 yr. 9 mos. at 6%? 
24, What is the interest of $80 for 4 yrs. 4 mos. at 8%? 
25. What is the interest of $600 for 5 mos. at 8%? 


SueGeEstion.— Find interest for 1 year, then for 1 month, then 


for 5 months. 


26. Find interest of $120 for 7 mos. at 4%. 
27. Find interest of $360 for 11 mos. at 9%. 


WRITTEN. 


28. Find interest of $240 for 3 years 7 months at 87. 


PROCEssS. 
12 $240 
.O8 
43 
$68.80 


Sotution I.—Interest for 1 year=$240x.08= 
$19.20. Interest for 375 yrs.=$19.20374s5= $68.80. 

SoLution IJ.—Express, in form for cancellation, 
the interest for 1 year, $240.08. 

Express the division of this product by 12, which 
is the interest for 1 month. Express the multi- 
plication. of this quotient by 45, the number of 


months (3 yrs.+7 mos.=48 mos.). 
By cancellation, the result is $68.80. 


206 ELEMENTARY ARITHMETIC. 


29. Find interest of $360 for 2 yrs. 5 mos. at 6%. 
30, Find interest of $600 for 3 yrs. 11 mos. at 8%. 
31. Find interest of $720 for 1 yr. 9 mos. at 8%. 
o2. Find interest of $480 for 3 yrs. 6 mos. at 10%. 
33. Kind interest of $90 for 4 yrs. 5 mos. at 6%. 
o4,. Find interest of $72 for 5 yrs. 3 mos. at 8%. 
35. Find interest of- $418 for 18 days at 6%. 


PROCEssS. SoLutTion.— Express, in form for cancellation, 
the interest for 1 year, $480.06. . 


*) o 
Ie $480 Express the division of this product by 12, which 
30 .06 ~ Neorg: ; ae 
18 result is the interest for 1 month. 


Express the division of this quotient by 30, which 
$1.44 result is the interest for 1 day. 
Express the multiplication of this quotient by 18, 
which result is the interest for 18 days. 
By cancellation, the result is $1.44. 


36. Find interest of $300 for 20 days at 8%. 

37. Find interest of $450 for 19 days at 10%. 

38. Find interest of $1200 for 16 days at 72. 

39. Find interest of $1500 for 25 days at 9%. 

40. Find interest of $840 for 7 months 21 days at 6%. 


PROCESS. 


12 840 
$ . SoLutTron.—Consider the 21 days as .7 of a month. 
yy Proceed as in example 28. 


$32.34 


41, Find interest of $660 for 3 mos. 18 days at 6%. 

42, Find interest of $340 for 5 mos. 24 days at 8%. 

43, Yind interest of $800 for 9 mos. 15 days at 47. 

44, Find interest of $140 for 7 mos. 12 days at 57. 

4, Find interest of $600 for 11 mos. 21 days at 7%. 
46, Find interest of $500 for 6 mos. 27 days at 10%. 
47. Find interest of $600 for 5 yrs. 3 mos. 11 days, 

at 5%: 


PERCENTAGE. 207 


PROCESS. 

12 | $600. Sotutron.—Interest on. $600 for 5 yrs.=$600x 

30 05 05 X5=$150. 3 mos+11 days=101 days. Find 
101 interest for 101 days by cancellation, as in example 


35, $8.414. 
$150-+-$8.41$= $158.41. 


Find interest of 

48. $960 for 2 yrs. 8 mos. 7 days, at 10%. 
49, $1440 for 3 yrs. 5 mos. 15 days, at 9%. 
50. $60 for 2 yrs. 3 mos. 20 days, at 742. 


Tap =1 4. 


SUGGESTION. ; 

51. $1209 for 4 yrs.-6 mos. 5 days, at 442. 

O2. $195 for 8 yrs. 1 mo. 3 days, at 64%. 

53. $180.75 fer 1 yr. 2 mos. 10 days, at 6%. 

54. $47.60 tor 2 yrs. 3 mos. 18 days, at 72. 

05. Find the amount of $325.50, for 4 yrs. 6 mos. 15 
days, at 6%. 

56. Find the amount of $480.90 for 3 yrs. 9 mos. 10 
days, at 8%. 

57. I bought 2 lot for $800, and after 2 yrs. 6 mos. I sold 
it for $1200. What was the gain, if money was worth 10%? 

58. What amount of money will pay a note of $500, 
due in 3 yrs. 6 mos. 20 days, at 10%? 

59. I loaned Mr. Jones $800, which he returned in 8 
mos. 10 days, with 8% interest. What did he pay me? 

60. What is the annual income from an investment of 


$10000, which pays 83%? 


REVIEW QUESTIONS. 


Define interest. Define principal. Define rate of interest. Define 
amount. What part of the principal is the interest for one year? 


208 


MISCELLANEOUS EXERCISES 


ELEMENTARY ARITHMETIC. 


IN PERCENTAGE. 


A. Find commission, or premium, or gain or loss, on 
numbers in first column, at 1%, 4%, 4%, 3% 14%, 22%, 64%, 81%. 
B. Find interest on each of the following principals for 


the time indicated, at 4%, 44%, 5%, 6%, 


Les 
18d. 

9d. 
21d. 
20d. 

dd. 
12d. 
27d. 


1.—$458.75 ly. 6m. 
2.—$680.50 2y. Im. 
3.—$320.82 3y. 2m. 
4.—$500.00 ly. 9m. 
i 1 DO.o1O 8m. 
6.—$873.40 dy. 2m. 
7.—$157.25 3y. 5m. 
8.—8$ 32.30 ly. 4m. 
9.—$750.00 3y. 8m. 
10.—$324.125 2y. 5m. 
11.—$862.17 2y. 9m. 
1 —$920.09 ly. 10m. 
13.— $382.11. 3y. 6m. 
14.—$809.05 2y. 7m. 
15.—$395.60 ly. 10m. 


6d. 
E heie 
Bd: 
19d. 
21d. 
12d: 


16.—$428.40 
17.—%600.00 
18.—$325.15 
19.—$216.80 
20.—$824.60 
21.—$324.20 
22.—$ 80.50 
23.—$192.60 
24.— $455.24 


25).—$128.125 


26.— 


$840.79 


27.—$224.375 2y. 


28.—%$800.00 
29.—$925.06 
30.—$708.20 


Lye eom 
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OY. MDs 
2y. dm. 
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ly. 3m 
dy. om 
9m 

vy. Vm 
10y. 6m. 
ay. 8m 


é, 74%, 8%, 9%, 108. 
Bey: 


6d. 


. 24d 


C. Find numbers of which the numbers below are 1% 


14%, 2%, 241%, 24%, 3%, 4%, 5%, 124%, 1624. 


D. Find what rate per cent each of the One Ee num- 


bers is of the succeeding number. 


1.— 40 
2.— 60 
d.— 64 
4.— 84 
D.— 96 
6.—132 
¢.—240 
— 276 
9.—300 
10.—312 
‘11.—342 
12.—426 
13.—405 
14.—414 
15.—510 


16.— 52d 

1.—620 
18.—720 
195— 3652 
20.—160 
le 
22.2—921 
A3.—3U05 
7 EE), By 
R0e—oO00 
26:—-piu 
27.—B891 
28.—927 
29.—800 


30.—639 


31.—711 
32.—425 
33.—632 
d4.—375 
39.—900 
36.—432 
37.—723 
38.—)22 
39.—7395 
40.—640 
41.—8382 
42.—729 
43,—324 
44.—108 
45,—216 


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